Graph Controllability Classes for the Laplacian Leader-Follower Dynamics
Cesar O. Aguilar and Bahman Gharesifard
Abstract—In this paper, we consider the problem of obtain- resulted in an extensive literature on controllability properties ing graph-theoretic characterizations of controllability for the for this class of control systems, see of example [7], [8], [9], Laplacian-based leader-follower dynamics. Our developments [10], [11], [12]. Specifically, starting with a Laplacian-based rely on the notion of graph controllability classes, namely, the classes of essentially controllable, completely uncontrollable, and dynamics, a subset of the agents are classified as leaders and conditionally controllable graphs. In addition to the topology act as control agents that can change the dynamics of the net- of the underlying graph, the controllability classes rely on the work. The remaining agents, called the followers, are indirectly specification of the control vectors; our particular focus is on controlled by the leaders via the connectivity of the network. the set of binary control vectors. The choice of binary control Most of the effort has gone to characterizing graph-theoretic vectors is naturally adapted to the Laplacian dynamics, as it captures the case when the controller is unable to distinguish conditions under which such systems are uncontrollable. between the followers and, moreover, controllability properties are invariant under binary complements. We prove that the class of essentially controllable graphs is a strict subset of the class of A. Literature review asymmetric graphs and provide numerical results that suggests The controllability of leader-follower network dynamics was that the ratio of essentially controllable graphs to asymmetric graphs increases as the number of vertices increases. Although first considered in [7], where a characterization of controlla- graph symmetries play an important role in graph-theoretic bility using spectral analysis of the system matrix was given. characterizations of controllability, we provide an explicit class Using a graph-theoretic approach, in [8] it was shown that of asymmetric graphs that are completely uncontrollable, namely for a single leader agent, symmetries present in the network the class of block graphs of Steiner triple systems. We prove that preserve the leader’s neighbors results in uncontrollabil- that for graphs on 4 and 5 vertices, a repeated Laplacian eigenvalue is a necessary condition for complete uncontrollability ity. Moreover, in the case of multiple leaders, a necessary but, however, show through explicit examples that for 8 and condition for controllability was given using equitable graph 9 vertices, a repeated eigenvalue is not necessary for complete partitions. In [9], it is shown that connectivity of the network is uncontrollability. For the case of conditional controllability, we necessary for controllability and two uncontrollable network give an easily checkable necessary condition that identifies a topologies are characterized. In [10], various sufficient and class of homogeneous binary control vectors that result in a two-dimensional controllable subspace. Finally, we give a lower- necessary conditions for controllability are given for a network bound on the rank of the controllability matrix for the Laplacian tree topology. In [11], sufficient and necessary conditions dynamics. Several constructive examples demonstrate our results. for controllability (and observability) of multi-input Laplacian dynamics for path and cycle network topologies are given in terms of modular arithmetic relations. In [12], a comprehensive study was undertaken of the controllability (and observability) I.INTRODUCTION properties of grid graphs. In particular, necessary and sufficient Many modern engineering and economic systems con- conditions are given that characterize the set of nodes that sist of collections of smaller subsystems interconnected to result in controllability. each other over a communication network. Examples include The controllability problem for graphs has received interest distributed energy resources, oscillator synchronization, dis- outside the control community. In [13], [14], the adjacency ma- tributed robotic networks [2], [3], [4], and also cascades of trix is used instead of the Laplacian matrix to study the graph information and opinions in social networks [5]. Due to the controllability problem. Explicitly, a “controllable graph” in large number of applications, in recent years there has been a [13], [14] is a graph whose adjacency matrix has distinct surge of activity within the control theory community to under- eigenvalues and no eigenvector of the adjacency matrix is stand how the network structure of multi-agent systems affects orthogonal to the all ones vector. Hence, the work in [13], [14] the fundamental properties of controllability and stabilizability. investigates controllability when all of the nodes are chosen With regards to controllability, a graph theoretic character- as leaders and the system matrix is given by the adjacency ization of linear networked control systems has been fully matrix. A similar approach is taken in [15], but now one is developed in [6] via the notion of structural controllability. allowed to control possibly only a subset of the nodes. We note In spite of this characterization, the recent attention within the that when the system matrix is the Laplacian instead of the control community on the so-called Laplacian dynamics has adjacency matrix, all graphs of order n ≥ 2 are uncontrollable according to the definition of “controllable graphs” given in Cesar O. Aguilar is with the Department of Mathematics, California State University, Bakersfield, CA 93311, USA [email protected] [13], [14], [15]. This follows from the well-known fact that and Bahman Gharesifard is with the Department of Mathematics the all ones vector is an eigenvector of the Laplacian matrix, and Statistics, Queen’s University, Kingston, ON K7L 3N6, Canada, and therefore orthogonal to the other eigenvectors. [email protected]. An incomplete version of this paper is submitted for presentation at the IEEE Control and Decision Conference 2014 The line of research in [7], [8], [9], [10] takes the point as [1]. of view that the states of the leaders act as inputs to the follower agents and the dynamics of the leaders are ignored. distinct controllability classes for graphs from order n =2 to As a result, the controllability analysis is undertaken on a n = 9. Throughout the paper, several examples demonstrate reduced order system. On the other hand, the line of research the results. in [11], [12], [16], [17] takes the point of view that the leaders continue to follow the Laplacian-based dynamics and the C. Organization external controls on the leaders influence the entire network The remainder of this paper is organized as follows. In through the interaction of the leaders with the followers. It is, Section II we establish some notation and present necessary however, an easy exercise to show that the former approach is definitions from graph theory along with a result on the a special case of the latter (see Remark 3.1). In this paper, our linear controllability for diagonalizable system matrices. In approach is more closely aligned with [11], [12], [16], [17]. Section III, after establishing some preliminary results, we discuss the motivation of this paper as it relates to the B. Statement of contributions existing literature on graph symmetries and uncontrollability of networked systems, and then introduce our graph control- The contributions of this paper are the following. To bet- lability classes. Section IV contain our main results. Finally, ter understand the role of topological graph obstructions to in Section V, we make concluding remarks and discuss ideas controllability for Laplacian-based leader-follower systems, for future work. we introduce graph controllability classes, namely, essentially controllable graphs, completely uncontrollable graphs, and II. PRELIMINARIES conditionally controllable graphs. These definitions rely on the The set of natural numbers is denoted by N and we set specification of the control vectors and we focus primarily on N N the case of binary control vectors. We show that with this 0 = {0} ∪ . Matrices will be denoted using upper case bold letters such as A, F, L, and vectors using lower case choice of control vectors, controllability is invariant under x b u A binary complements for Laplacian-based leader-follower dy- bold letters such as , , . The transpose of is denoted AT namics. by . The cardinality of a finite set S is denoted by |S|. If then the complement of in is denoted by . The As our first result on graph controllability classes, we prove S ⊂ R S R R\S standard basis vectors in Rn are denoted by e e e . that none of the essentially controllable graphs contain a non- 1, 2,..., n Finally, given u v Rn, we write that u v if u and identity graph automorphism, i.e., all such graphs are asym- , ∈ ⊥ v are orthogonal in the standard inner product of Rn, i.e., metric. As a by-product, the so-called minimal controllability vT u uT v . More generally, if Rn, we write problems [18] are solvable for this class. We also provide = = 0 W ⊂ u if u w for each w . numerical results that suggest that the ratio of essentially ⊥ W ⊥ ∈ W controllable graphs to asymmetric graphs tends to one as the number of vertices increases. A. Graph theory We then provide an explicit class of graphs, namely the Our notation from graph theory is standard and follows the block graphs of Steiner triple systems, that are asymmetric yet notation in [20], [21]. By a graph we mean a pair G = (V, E) completely uncontrollable, and thus showing that essentially consisting of a finite vertex set V and an edge set E ⊆ [V]2 := controllable graphs form a strict subset of asymmetric graphs. {{v, w} | v, w ∈ V}. We consider only simple graphs, i.e., Although symmetry plays an important role in graph-theoretic unweighted, undirected, with no loops or multiple edges. The characterizations of controllability [8], this result and the fact order of the graph G is the cardinality of its vertex set V. The that asymmetry is typical in finite graphs [19], suggests that neighbors of v ∈ V is the set Nv := {w ∈V|{v, w}∈E} the current focus in the literature on characterizing graph and the degree of v is dv := |Nv|. A path in G of length k is uncontrollability by identifying graph symmetries targets a a subgraph of G consisting of vertices {v0, v1,...,vk}⊂V narrow non-generic scenario. We prove that for connected and edges {{v0, v1}, {v1, v2},..., {vk 1, vk}}⊂E, where all − graphs with four or five vertices, a repeated eigenvalue is a the vi are distinct. For such a path, v0 and vk are called necessary condition for complete uncontrollability but show the terminal vertices. Given vertices u, v ∈ V, we define through explicit examples that for n ≥ 8 this condition is in the distance d (u, v) between u and v as the length of a general not necessary. As a by-product of our results, we give shortest path whoseG terminal vertices are u and v. A graph a sufficient condition for complete uncontrollability in terms G is connected if there is a path between any pair of vertices. of the eigenvectors of the Laplacian matrix and construct a Henceforth, when not explicitly stated, we fix an ordering on class of non-regular completely uncontrollable graphs. the vertex set V and thus, without loss of generality, we take We then provide a sufficient condition for conditional con- V = {1,...,n}, where n is the order of G. The adjacency trollability. Specifically, we identify a class of binary vectors matrix of G is the n × n matrix A defined as Aij = 1 if that we call homogeneous that result in a two-dimensional {i, j}∈E and Aij = 0 otherwise, where Aij denotes the controllable subspace. As an example, we show that the 3- entry of A in the ith row and jth column. We note that if k regular asymmetric Frucht graph on twelve vertices possess r = d (i, j), with i 6= j, then (A )ij = 0 for all 0 ≤ k < r G r these homogeneous binary control vectors. We then provide and (A )ij 6=0. a lower-bound on the rank of the controllability matrix for We denote by D the degree matrix of G, i.e., the diagonal the Laplacian based leader-follower system and discuss its matrix whose ith diagonal entry is di. The Laplacian matrix implications in the controllability of path graphs. Finally, of G is given by we end the paper with numerical results enumerating the L = D − A. The Laplacian matrix L is symmetric and positive semi- system definite, and thus the eigenvalues of L can be ordered λ1 ≤ x˙ (t)= −Lx(t), T λ2 ≤ ··· ≤ λn. The ones vector 1n := [1 1 ··· 1] is where x ∈ Rn, t ∈ R, and L is the Laplacian matrix of G. an eigenvector of L with eigenvalue λ1 = 0, and if G is Suppose that a nonempty subset of the vertices V ⊂V are connected then λ1 = 0 is a simple eigenvalue of L. We actuated by a single control u : [0, ∞) → R and consider assume throughout that G is connected so that 0 < λ2. For our purposes, by the eigenvalues (eigenvectors) of a graph G the resulting single-input linear control system. Explicie tly, let b T n we mean the eigenvalues (eigenvectors) of its Laplacian matrix = [b1 b2 ··· bn] ∈{0, 1} be the binary vector such that L. V = Vb := {i ∈ V | bi = 1}, and consider the single-input A mapping ϕ : V→V is an automorphism of G if it is a linear control system bijection and implies that . The e {i, j}∈E {ϕ(i), ϕ(j)}∈E x˙ (t)= −Lx(t)+ bu(t). (1) order of an automorphism ϕ is the smallest positive integer k such that the k-fold composition of ϕ with itself is the The vertices Vb are seen as control or leader nodes and influ- identity automorphism. An automorphism ϕ of G induces a ence the remaining follower nodes V\Vb through the control n linear transformation on R , denoted by Pϕ or just P when signal u(·) and the connectivity of the network. A motivation ϕ is understood, whose matrix representation in the standard for the set of binary control vectors is that it captures the basis is a permutation matrix, i.e., as a linear mapping ϕ scenario of when an external agent connected to the nodes acts as a permutation on the standard basis {e1,..., en} Vb is unable to distinguish between its followers. Hence, all of Rn. It is well-known that ϕ is an automorphism of G the followers receive the same control input from the leader. if and only if PA = AP. Moreover, an automorphism P The reason for choosing the Laplacian dynamics (1) is that preserves degree of vertices, and therefore di = dϕ(i) for it serves as a benchmark problem for studying distributed every i ∈ {1, 2,...,n}, i.e., PD = DP. It follows that an control systems. The problem is also of independent theoretical automorphism P of G also satisfies PL = LP. interest because it reveals useful information about the set of A graph is called k-regular if all its vertices have degree eigenvectors of the Laplacian matrix of a graph [15]. k ∈ N.A k-regular graph G = (V, E) is called strongly regular From a controls design perspective, it would of course be if there exists λ, µ ∈ N such that desirable to select the leader nodes so that the pair (L, b) is b 1 i) |Nv ∩ Nu| = λ, for every v ∈V and every u ∈ Nv, and controllable. First, note that choosing = n results in a L b L1 0 ii) |Nv ∩ Nu| = µ, for every v ∈V and every u∈ / Nv. controllable pair ( , ) if and only if n =1 since n = n. It is known that strongly regular graphs have exactly three More generally, a direct application of Proposition 2.1(ii) for Laplacian eigenvalues [22]. A strongly regular graph will be the Laplacian dynamics yields the following result. denoted by SRG(n,k,λ,µ). Corollary 3.1 ([7]): (Necessary and sufficient condition for controllability of Laplacian dynamics): Consider the controlled B. Diagonalizability and linear controllability Laplacian dynamics (1) with b ∈ Rn and assume that L has n n n L b Given a matrix F ∈ R × and vector b ∈ R , we no repeated eigenvalues. Then the pair ( , ) is controllable denote by hF; bi the smallest F-invariant subspace containing if and only if b is not orthogonal to any eigenvector of L. k b. It is well-known that hF; bi = span{F b | k ∈ N0}, Although Corollary 3.1 provides a general necessary and and that if dimhF; bi = k + 1 then {b, Fb,..., Fkb} is sufficient condition for controllability in terms of the graph a basis for hF; bi. The pair (F, b) is called controllable Laplacian eigenvectors, the problem that we consider is in if dimhF; bi = n. The following result characterizes the obtaining controllability conditions in terms of the topological controllability of single-input linear systems (F, b) when F structure of the graph. Graph-theoretic characterizations of is diagonalizable. controllability for leader-follower multi-agent systems was Proposition 2.1: (Controllability and eigenvalue multiplic- first considered in [8] in terms of the automorphism group n n b n ity): Let F ∈ R × be diagonalizable. of a graph. Following [8], we say that ∈ {0, 1} is (i) For any open set B ⊂ Rn, the pair (F, b) is uncontrol- leader symmetric if there exists a non-trivial automorphism lable for every b ∈ B if and only if F has a repeated ϕ : V→V of G that leaves the leader nodes Vb invariant, i.e., eigenvalue. Pϕ(b) = b. It is straightforward to verity that the definition (ii) Suppose that F has distinct eigenvalues and let U of leader symmetry given in [8] is equivalent to the one be a matrix whose columns are linearly independent given here. The following result of [8] links leader symmetry eigenvectors of F. If b ∈ Rn then the dimension of and uncontrollability (a short alternative proof is given in the hF; bi is equal to the number of nonzero components of Appendix). 1 v = U− b. In particular, (F, b) is controllable if and Proposition 3.1: (Leader symmetry and uncontrollability): only if no component of v is zero. Consider the controlled Laplacian dynamics (1) with b ∈ The proof of (i) follows from the properties of the determinant {0, 1}n. If b is leader symmetric then (L, b) is uncontrollable. and for the proof of (ii) see for instance [7]. As shown in [8, Proposition 5.9], leader symmetry is not a necessary condition for uncontrollability. Figure 1(a) displays III. PROBLEM STATEMENT AND GRAPH the graph on n =6 vertices that is used in [8] to show this fact. CONTROLLABILITY CLASSES Unfortunately, this example is not illuminating in the quest for Let G = (V, E) be a graph with vertex set V = obtaining graph-theoretic characterizations of controllability {1, 2,...,n}. The Laplacian dynamics on G is the linear because the leader nodes are chosen so that b = 1n, i.e., Remark 3.1: Let us describe the approach taken in [8] and how it relates to ours. We note that our approach is also adopted in [11], [12]. In [8], one begins with a Laplacian- (a) (b) (c) based dynamics x˙ = −Lx, selects a leader node, say i ∈ Fig. 1. (a) The example of [8], (b) an asymmetric graph on n = 6 vertices {1, 2,...,n}, and considers the reduced system of followers (n 1) (n 1) having 14 binary vectors b resulting in uncontrollable Laplacian dynamics, actuated by node i. Explicitly, let Lf ∈ R − × − be the and (c) a graph for which any binary vector b results in uncontrollability. matrix obtained by deleting the ith row and ith column of L, and let b ∈ Rn 1 be the column vector obtained by removing every node is actuated. As remarked above, unless , f − n = 1 the ith entry of the ith column of L. The reduced system of this choice results in uncontrollability regardless of the graph followers considered in [8] is z˙ = −L z − b u. The system topology. Interestingly, the only control vectors b resulting in f f (Lf , bf ) is controllable if and only if (L, ei) is controllable. uncontrollability for the graph in Figure 1(a) are the trivial Indeed, the dynamic extension ones, i.e. b = 0n or b = 1n. In view of the fact that asymmetry is typical in finite graphs [19], it is natural then to z˙ = −Lf z − bf ξ, ask what graph-theoretic obstructions to controllability exist ξ˙ = v, other than symmetry. For example, consider the asymmetric L b graph on n = 6 vertices displayed in Figure 1(b). Of the is controllable if and only if ( f , f ) is controllable. Letting n bT z 2 − 2=62 non-trivial choices of b, there are 14 that result v = − f − diξ + u, we see that the dynamic extension in uncontrollability, namely: is feedback equivalent to (L, ei). Hence, in relation to the problem we consider in this paper, the approach in [8] is T T b1 = [1 1 1 0 0 0] , b2 = [0 0 0 1 1 1] , concerned with the controllability of (L, b) in the restricted T T b e e e n b3 = [1 1 0 1 0 0] , b4 = [0 0 1 0 1 1] , case that ∈ { 1, 2,..., n}⊂{0, 1} . We note that the T T graph in Figure 1(b) is such that (L, ei) is controllable for b5 = [0 1 1 1 0 0] , b6 = [1 0 0 0 1 1] , every ei ∈ {e1,..., en}, yet as shown in the example, fails b T b T 7 = [0 1 0 0 1 0] , 8 = [1 0 1 1 0 1] , (2) to be controllable for some b ∈{0, 1}n. T T b9 = [1 0 1 0 1 0] , b10 = [0 1 0 1 0 1] , T T IV. GRAPH-THEORETIC CHARACTERIZATIONS OF b11 = [1 0 0 1 1 0] , b12 = [0 1 1 0 0 1] , CONTROLLABILITYCLASSES b = [0 0 1 1 1 0]T , b = [1 1 0 0 0 1]T . 13 14 Before we state our main results, we provide a useful The control vectors b7 and b8 result in a two-dimensional property of controllability under binary control vectors. The controllable subspace, while the other control vectors all result astute reader may have noticed that the control vectors (2) that in a five-dimensional controllable subspace. On the other hand, result in uncontrollability for the graph in Figure 1(b) come in for the graph on n =6 vertices displayed in Figure 1(c), any complementary pairs. To be more precise, given b ∈{0, 1}n choice of b ∈{0, 1}n results in uncontrollability. Clearly, the we let graph displayed in Figure 1(c) has a non-trivial symmetry but b = 1n − b symmetry plays no role in the lack of controllability for every b n be the complement of . As a further piece of notation, for control vector b ∈ {0, 1} . In fact, as we will show, there n b ∈{0, 1}n we let kbk = b be the number of nonzero exist asymmetric graphs such that no matter the choice of 1 i=1 i elements of b. With this notation we have the following result. b ∈{0, 1}n the pair (L, b) is uncontrollable. Proposition 4.1: (ControllabilityP and binary complements): Our previous discussion naturally leads to the definition of Let n ≥ 2 and consider the controlled Laplacian dynamics the following three graph controllability classes. (1) with b ∈ {0, 1}n. Then the pair (L, b) is controllable Definition 3.1: (Graph controllability classes): Let G be a if and only if the pair (L, b) is controllable. Specifically, if connected graph with Laplacian matrix L and let B ⊂ Rn be b ∈{/ 1 , 0 } then dimhL; bi = dimhL; bi. a nonempty set. Then G is called n n Proof: If L has repeated eigenvalues, then (L, c) is un- (i) essentially controllable on B if (L, b) is controllable for controllable for every c ∈ Rn, and the claim follows trivially. every b ∈ B\ ker(L); Hence, assume that L has distinct eigenvalues. Let U be an (ii) completely uncontrollable on B if (L, b) is uncontrollable orthogonal matrix consisting of unit norm eigenvectors of L for every b ∈ B; and let the first column of U be the eigenvector u = 1 1 . (iii) conditionally controllable on B, if it is neither essentially 1 √n n b n 1 0 v UT b T controllable nor completely uncontrollable on B. Let ∈{0, 1} \{ n, n}, let = = [v1 ··· vn] , and v UT b T In this paper we are mainly concerned with controllability let = = [¯v1 ··· v¯n] . Then classes on the control set B = {0, 1}n. Hence, when not v n e v = √ 1 − . (3) explicitly stated, we simply call a graph G essentially control- n lable (conditionally controllable, or completely uncontrollable) From (3) we see that v¯i = −vi for all i = 2,...,n. Now, T b 1 u b k k b 0 if G is essentially controllable (conditionally controllable, or v1 = 1 = √n and thus v1 6= 0 because 6= n. n n b 1 completely controllable) on {0, 1} . Hence, according to Def- −k k On the other hand, v¯1 = √n and thus v¯1 6= 0 because inition 3.1, the graph in Figure1(a) is essentially controllable, b 6= 1n. This proves that v and v have the same number of the graph in Figure1(b) is conditionally controllable, and the nonzero components provided b ∈/ {1n, 0n}. Therefore, by graph in Figure1(c) is completely uncontrollable. Proposition 2.1(ii), we have that dimhL; bi = dimhL; bi. For computational purposes, it is worth mentioning the following immediate consequence of the previous result. Corollary 4.1: (Uncontrollable subset has even cardinal- ity): If n ≥ 2 then the cardinality of the set of {b ∈ n {0, 1} | (L, b) is uncontrollable} is always even. (a) (b)
Fig. 3. Essentially controllable graphs of order n = 8 (a), and n = 11 (b). A. Essentially controllable graphs The class of essentially controllable graphs are interesting In this section, we give a necessary condition for essential for various reasons. First, this class is important from a design controllability. The condition depends on the following auxil- perspective because, except for the trivial control vectors 0 iary result. n and 1n, controllability is independent of the subset of nodes Lemma 4.1: (Order of non-identity automorphisms [20]): If that receive the control inputs. This is useful when this is un- L all of the eigenvalues of are simple then every non-identity known a priori, e.g., when the control inputs are broadcasted. automorphism of G has order two. Another important fact about essentially controllable graphs is L Proof: The proof of the claim when is replaced by the that the so-called minimal controllability problem is solvable A adjacency matrix is given in [20, Theorem 15.4]. However, [18] for these graphs. Following [18], let B ⊂ Rn and consider L P the proof for the case of is identical because if is an the dynamics (1) for fixed b ∈ B. We say that (1) is minimally P automorphism of G then commutes with both the adjacency controllable if b has the fewest number of nonzero entries A D P matrix and the degree matrix , and therefore also among all vectors b˜ ∈ B such that (L, b˜) is controllable. L commutes with . It is shown in [18] that it is in general intractable to even Using the previous result, the following necessary condition approximate the number of zeros in the vector b that leads to for essential controllability is straightforward. minimal controllability. Nevertheless, given that the class of Proposition 4.2: (Essentially controllable graphs are asym- essentially controllable graphs are controllable using any non- metric): Let n ≥ 3. An essentially controllable graph on trivial vector in {0, 1}n, the minimal controllability problem n {0, 1} is asymmetric. is solvable for (1) on all essentially controllable graphs, and Proof: Let G be an essentially controllable graph on the sparsest b ∈{0, 1}n has (n − 1) nonzero entries. n L {0, 1} . Then necessarily must have distinct eigenvalues and We are not aware of any algorithm producing essentially therefore, by Lemma 4.1, every non-identity automorphism of controllable graphs. Given that these graphs constitute a strict G has order two. Assume by contradiction that G has a non- subset of asymmetric graphs, and that it is NP-hard to verify if P trivial automorphism group and let be a permutation matrix a graph has non-trivial automorphisms [23], it is unclear if the representing a non-identity automorphism of G. Then there problem of generating essentially controllable graphs of order e e exists two distinct standard basis vectors i and j such that n is computationally feasible. Another interesting problem Pe e Pe e b e e i = j and j = i. Put = i + j . We note that since is to investigate how the number of essentially controllable b 1 b n ≥ 3 we have that 6= n. Now, is clearly invariant under graphs grows, within the class of asymmetric graphs, with the P Pb b b , i.e., = . Thus, is leader symmetric and therefore, number of vertices (see Table I). by Proposition 3.1, (L, b) is uncontrollable, a contradiction. This completes the proof. B. Completely uncontrollable graphs According to Proposition 4.2, and since any asymmetric graph has at least six vertices [19], any essentially controllable In this section, we study the class of completely uncon- graph has also at least six vertices. The condition given in trollable graphs. Our first result shows that complete uncon- Proposition 4.2 is, however, clearly only necessary; the graph trollability is not a consequence of graph symmetries. To of Figure 1(b) is an example of an asymmetric graph with six the best of our knowledge, this important fact is overlooked nodes that is not essentially controllable. in the literature on network controllability primarily because Example 4.1: (Essentially controllable graphs): Exactly most existing results seek graph-theoretic characterizations of four of the eight asymmetric graphs on six vertices are uncontrollability via graph symmetries. To state our first result, essentially controllable; these graphs are shown in Figure 2. we need to introduce a subclass of strongly regular graphs called the block graphs of Steiner systems [24]. We begin with the following definition. Definition 4.1: (Steiner systems): Given three integers 2 ≤ t
Fig. 5. (a) and (b) show two completely uncontrollable graphs with n = 8 Put U = u1 u2 u3 u4 ··· un and let Λ = vertices and (c) shows a completely uncontrollable graphs with n = 9 vertices, diag(λ1,...,λn) be the corresponding diagonal matrix of all with distinct eigenvalues. eigenvalues of L, where λ1 = 0 and λ2,...,λn are in no Needless to say, the class of completely uncontrollable particular order. Now, since L = UΛUT , a straightforward graphs with distinct eigenvalues form a very special class of calculation shows that the upper left 4 × 3 submatrix of L is graphs and have the potential to shed light on new necessary 1 λ + 1 λ − 1 λ + 1 λ − 1 λ conditions for controllability and will be pursued in a future 2 2 4 4 2 2 4 4 4 4 paper. For now, we focus on obtaining conditions that imply 1 1 1 1 1 − 2 λ2 + 4 λ4 2 λ2 + 4 λ4 − 4 λ4 the existence of a repeated eigenvalue and consequently com- L4 3 = . × 1 1 1 1 plete uncontrollability. To that end, we introduce the following − λ4 − λ4 λ3 + λ4 4 4 2 4 definition. B B Rn − 1 λ − 1 λ − 1 λ + 1 λ Definition 4.3: ( -annihilators): Let ⊂ and let γ = 4 4 4 4 2 3 4 4 u u Rn be linearly independent. We say that is { 1,..., k}⊂ γ Since λ4 > 0, and the off diagonal entries of L are either a B-annihilator or that it annihilates B if for each b B ∈ 0 or −1, we obtain from the (1, 3) entry of L that λ4 = 4. there exists u that is orthogonal to b, that is, uT b . j ∈ γ j =0 Then, from the (1, 2) entry of L, either λ2 = 2 or λ2 = 4. The proof of Proposition 4.3 identifies a set of three vectors In the latter case, L has a repeated eigenvalue, which is a n that alone are {0, 1} -annihilators. contradiction, and therefore λ2 =2. Similarly, from the (4, 3) n Lemma 4.2: (A set of {0, 1} -annihilator vectors): Let n ≥ entry of L, the only possible cases are λ3 = 2 or λ3 = 4. v v v Rn 4 be a positive integer and let 1, 2, 3 ∈ be defined by In either case, L has a repeated eigenvalue, which leads to a contradiction. This completes the proof. v T 1 = 1 −1 0 0 0 ··· 0 , In Theorem 4.1, we showed the existence of (asymmetric) v T 2 = 0 0 1 −1 0 ··· 0 , (4) completely uncontrollable graphs which happened to be regu- T lar graphs. To end this section, we use Lemma 4.2 to construct v 1 1 −1 −1 0 ··· 0 3 = . a class of non-regular completely uncontrollable graphs. n Then {v1, v2, v3} is a {0, 1} -annihilator. Theorem 4.3: (Large uncontrollable graphs): For each n ≥ Proof: Any vector b ∈{0, 1}n having a zero in compo- 6, the set of graphs of order n that are not regular and are nents 1 through 4 is clearly orthogonal to v1 (and v2, and v3). completely uncontrollable is nonempty. Therefore, we need only consider the binary vectors having Proof: We prove the result by a direct construction. For possibly nonzero entries in components 1, 2, 3, and/or 4. There n =6, consider the graph in Figure 6(a). The Laplacian matrix 4 4 for this graph is are k=1 k = 15 possible cases: T i) if b = e1 or b = e2 then b v2 =0; 1 −10 0 0 0 P T ii) if b = e3 or b = e4 then b v1 =0; −1 5 −1 −1 −1 −1 b b bT v iii) if (1) = (2) = 1 then 1 =0; L 0 −1 3 −1 −1 0 b b bT v 6 = , iv) if (1) = (3) = 1 then 3 =0; 0 −1 −1 3 0 −1 b b bT v v) if (1) = (4) = 1 then 3 =0; 0 −1 −1 0 3 −1 vi) if b b then bT v ; (2) = (3) = 1 3 =0 0 −1 0 −1 −1 3 and a set of linearly independent eigenvectors of L are be used to identify subsets of other Ck’s. We begin with the following definition. u = 1 1T , 1 √6 6 Definition 4.4: (Homogeneous control vectors): Let G = T N b n u = 1 5 −1 −1 −1 −1 −1 , (V, E) be a graph and let α, β ∈ . We say that ∈{0, 1} is 2 √30 a (α, β)-homogeneous control vector for G if, for each i ∈Vb, 1 T u = 0 0 −1 0 0 1 , b b 3 √2 we have that α = |Ni ∩ V\V | and, for each j ∈ V\V , we T have that β = |N ∩Vb|. In other words, each leader node i is u = 1 0 0 0 1 −1 0 , j 4 √2 adjacent to α followers and each follower node j is adjacent u 1 T to β leaders. 5 = 2 0 0 1 −1 −1 1 , T Theorem 4.4: (Rank two control vectors): Let G be a graph u = 1 −401111 . 6 √20 and consider the controlled Laplacian dynamics (1), where b n 0 1 b After a permutation of the indices, we can apply Lemma 4.2 ∈ {0, 1} \{ n, n}. If is a (α, β)-homogeneous control 6 vector for G then b ∈C2. In fact, and conclude that the set {u3, u4, u5} is a {0, 1} -annihilator. Now let n ≥ 6 and extend the graph in Figure 6(a) to the graph L2b = (α + β)Lb G shown in Figure 6(b), where Gn 6 is any connected graph on n − 6 vertices. By construction,− the Laplacian of G can be and therefore Lb is an eigenvector of L with eigenvalue (α + β). Consequently, dimhL; bi =2. Conversely, if dimhL; bi = 6 Gn−6 2 then Lb is an eigenvector of L. 1 4 2 1 7 Proof: Consider the discrete linear system 2 x Lx 3 (k +1)= (k), 4 3 6 5 5 with initial condition x(0) = b ∈ {0, 1}n. Let x(k) = (a) (b) T x1(k), ··· , xn(k) denote the state vector at time k ∈ N Fig. 6. A {0, 1}n-annihilator graph with six vertices (a), and its extension 0. If i ∈ Vb then xi(0) = 1 and therefore xi(1) = n to a , -annihilator graph of any size (b). (x (0) − x (0)) = α. If j ∈ V\Vb then x (0) = 0 {0 1} ℓ i i ℓ j and∈N therefore . In other xj (1) = ℓ j (xj (0) − xℓ(0)) = −β decomposed as P ∈N L E words, 6 P L = , Lb x b b 1 b 1 T = (1) = α + β( − n) = (α + β) − β n. "E Ln 6# − Then, where Ln 6 denotes the Laplacian of the graph Gn 6 and − − E R6 (n 6) 2 ∈ × − is the matrix L b = x(2) = Lx(1) = (α + β)Lb − βL1n = (α + β)Lb e 0 0 E = − 1 n ··· n and this proves the first claim. Now, if dimhL; bi = 2 then L2b = c b + c Lb for some From the above decomposition of L, and noting that the first 0 1 c ,c ∈ R. Using the fact that L2b is orthogonal to 1 we entries of u , u , u are zero, it is not hard to see that u , u 0 1 n 3 4 5 3 4 immediately deduce that c = 0. Hence, L2b = c Lb, i.e., and u can be lifted to eigenvectors of L. Indeed, we have 0 1 5 Lb is an eigenvector of L. This ends the proof. that u L u u The following corollary is immediate. L j 6 j j = = λj . Corollary 4.2: Let be a connected graph on -vertices 0n 6 0n 6 0n 6 G n − − − u and suppose that n ≥ 3. If G has a (α, β)-homogeneous j Rn It is clear that the lifted eigenvectors 0n−6 ∈ , for j ∈ control vector then G is not essentially controllable. {3, 4, 5}, form a set of {0, 1}n annihilators. This ends the We give two examples that illustrate the previous result. proof. Example 4.5: (A rank two conditionally controllable asym- metric graph): Consider the asymmetric graph shown in Fig- b L b C. Conditionally controllable graphs ure 7. There are 2 binary vectors that result in dimh , i = 2, namely b = e2 + e5 and its binary complement b. For In this section we consider conditionally controllable the choice b, each control node (red nodes) is adjacent to graphs. The goal for this class would be to classify, in graph- α = 2 follower nodes (blue nodes) and each follower node b n theoretic terms, the set of control vectors ∈ {0, 1} such is adjacent to β = 1 control node. It can be verified that L b that dimh ; i = k, for each k ∈{0, 1, 2, 3,...,n}. In other L2b = (α + β)Lb =3Lb. • words, letting Example 4.6: (Frucht graph): As another example, consider n the Frucht graph, shown in Figure 8, which is an asymmetric Ck = {b ∈{0, 1} | dimhL; bi = k}, 3-regular graph on n = 12 vertices. There are 4 binary vectors n for k ∈ {0, 1, 2,...,n}, so that {0, 1} = C0 ∪C1 ∪C2 ∪ b that result in the controllable subspace of (L, b) having ···∪Cn and Ci ∩Cj = ∅ whenever i 6= j, we would like to dimension two, namely, b1 = e1+e5+e7+e12, b2 = e3+e7+ develop graph-theoretic conditions that fully characterizes Ck. e10, and their binary complements b1 and b2, respectively. To this end, the main result in this section is the identification The cases b1 and b1 are shown in Figure 8 and the cases of elements in C2. It is feasible that a similar technique can b2 and b2 are shown in Figure 9. For b1, each control node ✻
✻ The following result appears in [17] for the single-leader ✹
✹ case and in [16] for the multiple-leader case. To keep this ✺
✺ paper self-contained, we include its short proof.
✷ ✷
✶ ✶ ✸ ✸ Theorem 4.5: (A lower-bound on the rank of the control- lability matrix): Let G = (V, E) be a connected graph Fig. 7. Leader configuration b = e2 + e5 (left) and its binary complement b b¯ b and consider the Laplacian dynamics (1), where ∈ (right). For , each control node (red) is adjacent to 2 follower nodes (blue) n and each follower node is adjacent to 1 control node. {0, 1} \{1n, 0n}. Then
dimhL; bi≥ rb +1, (5)
(red nodes) is adjacent to α =2 follower nodes (blue nodes) where rb is the control radius of b. and each follower node is adjacent to β = 1 control node. It Proof: For a follower node j ∈ V\Vb let Kj := {i ∈ 2 can be verified that L b1 = (α + β)Lb1 = 3Lb1. For b2, Vb | d (i, j)= rj }. From Lemma 4.3, and linearity, it follows Gk each control node is adjacent to α = 3 follower nodes and that (L b)j =0 for all 0 ≤ k < rj and each follower node is adjacent to β = 1 control node. One r r r 2 (L j b) = (−1) j (A j ) . can verify that L b2 = (α + β)Lb2 =4Lb2. • j ij i Kj
X∈
✶✷ ✶✷ ✶ ✶ k
It is well-known that (A )ij is the number of walks from i to
✷ ✷ ✾
✾ j of length k [20, pg. 11]. Hence, by definition of rj , we have
✵ ✵
✶ ✶
✶✶ ✶✶ ✸ ✸ rj rj
(A )ij > 0 for all i ∈ Kj. This implies that (L b)j 6=0. It ✺
✺ follows that {b, Lb,..., Lrj b} is a linearly independent set
✽ ✽
✼ ✼
✹ ✹
of vectors. The claim follows by taking rj = rb. ✻ ✻ Using Theorem 4.3, we obtain an alternative proof of the following known fact about the controllability of a path graph. Fig. 8. Leader configuration b1 = e1 + e5 + e7 + e12 (left) and its binary Corollary 4.3: (Controllability of path graphs [7]): The path b¯ b complement 1 (right). For 1, each leader node (red) is connected to 2 graph is controllable when the leader node is chosen as follower nodes (blue) and each follower is connected to 1 leader node, and Pn vice-versa for the complement b¯1. one of the terminal nodes. Proof: Let G = Pn = {{1, 2,...,n}, {{1, 2}, {2, 3},..., {n − 1,n}} be the
b e ✶✷
✶✷ path graph of length . If then the control radius is ✶ ✶ n = n
b ✷
✷ r = d (1,n) = n − 1, and therefore by Theorem 4.5 we ✾
✾ G
✵ ✵ ✶
✶ must have that L b , i.e., L b is controllable. ✶ ✶
✶ ✶ dimh ; i = n ( , )
✸ ✸
✺ ✺
✽ ✽
✼ ✼ ✹
✹ D. Enumeration of Controllability Classes for Small Graphs ✻ ✻ In this section we provide numerical results on the cardinal- ity of the controllability classes. In Table I, we enumerate the Fig. 9. Leader configuration b2 = e3 + e7 + e10 (left) and its binary graph controllability classes for small connected graphs from complement b¯2 (right). For b2, each leader node (red) is connected to 3 follower nodes (blue) and each follower is connected to 1 leader nodes, and order n =2 through n =9. In the table, gn is the number of vice-versa for the complement b¯2. connected graphs, an is the number of asymmetric connected graphs, en is the number of essentially controllable graphs, To end this section, we provide a lower-bound for un is the number of completely uncontrollable graphs, and cn dimhL, bi. The result depends on the following whose proof is the number of conditionally controllable graphs, where n is is found in the Appendix, see also [16]. the order of the graph. Lemma 4.3: (An auxiliary result on powers of the Laplacian matrix): Let G = (V, E) be a connected graph with vertex set n gn an en un cn V = {1, 2,...,n}, and let r ∈ N. Then for i, j ∈V such that 21 0 10 0 k 32 0 01 1 r ≤ d (i, j) we have that (L )ij =0, for all 0 ≤ k < r, and G 46 0 04 2 5 21 0 0 11 10 r r r (L )ij = (−1) (A )ij . 6 112 8 4 59 49 7 853 144 84 264 505 To state our lower bound, we need a further piece of 8 11117 3552 1992 2764 6361 9 261080 131452 94084 29750 137246 notation. Given a follower node j ∈ V\Vb, we denote by rj the minimum distance of j to the set of control nodes, that TABLE I is, ENUMERATIONOFCONTROLLABILITYCLASSESFORSMALLGRAPHS rj = min d (i, j). Values of the sequences and can be found in [31]. We i b G gn an ∈V used Maple’s Graph Theory package to generate the adjacency We define the control diameter of b by matrices of all connected graphs on 2 ≤ n ≤ 9 vertices. The data in Table I suggests that the ratio en/an is growing rb = max rj . j b with n. It is an interesting problem to investigate if almost all ∈V\V asymmetric graphs are essentially controllable on {0, 1}n as [10] Z. Ji, H. Lin, and H. Yu, “Leaders in multi-agent controllability under n → ∞. consensus algorithm and tree topology,” Systems Control Letters, vol. 61, pp. 918–925, 2012. [11] G. Parlangeli and G. Notarstefano, “On the reachability and observability of path and cycle graphs,” IEEE Trans. Autom. Control, vol. 57, no. 3, ONCLUSION AND UTURE ORK V. C F W pp. 743–748, 2012. We have considered the controllability problem for the [12] G. Notarstefano and G. Parlangeli, “Controllability and observability of grid graphs via reduction of symmetries,” IEEE Trans. Autom. Control, Laplacian-based leader-follower dynamics. We introduced the vol. 58, no. 7, pp. 1719–1731, 2013. class of essentially controllable, completely uncontrollable, [13] D. Cvetkovi´c, P. Rowlinson, Z. Stani´c, and M.-G. Yoon, “Controllable and conditionally controllable graphs. We proved that the set graphs,” Bull. Acad. Serbe Sci. Arts. Classe Sci. Math. Natur. Sci. Math., vol. 140, no. 36, pp. 81–88, 2011. of essentially controllable graphs is strict subset of the set of [14] D. Cvetkovi´c, P. Rowlinson, Z. Stani´c, and M.-G. Yoon, “Controllable asymmetric graphs and classified all essentially controllable graphs with least eigenvalue at least −2,” Appl. Anal. Discrete Math., graphs on six vertices. We provided a class of asymmetric vol. 5, pp. 165–175, 2011. [15] C. Godsil, “Controllable subsets in graphs,” Ann. Comb., vol. 16, completely uncontrollable graphs, namely the block graphs of pp. 733–744, 2012. Steiner triple systems. We proved that for connected graphs [16] A. Yazicioglu, W. Abbas, and M. Egerstedt, “A tight lower bound on with four or five vertices, having a repeated eigenvalue fully the controllability of networks with multiple leaders,” in IEEE Conf. on Decision and Control, pp. 1978–1983, 2012. characterizes complete uncontrollability. We gave a suffi- [17] S. Zhang, M. Camlibel, and M. Cao, “Controllability of diffusively- cient condition for complete uncontrollability in terms of the coupled multi-agent systems with general and distance regular coupling eigenvectors of the Laplacian matrix, which also leads to topologies,” in IEEE Conf. on Decision and Control, pp. 759–764, 2011. [18] A. Olshevsky, “Minimal controllability problems,” preprint, 2014. repeated eigenvalues. We have also shown the existence of http://arxiv.org/abs/1304.3071. completely uncontrollable graphs with distinct eigenvalues for [19] P. Erd¨os and A. R´eyni, “Asymmetric graphs,” Acta Math. Acad. Sci. graphs with eight vertices and higher. We identified a class Hungary, vol. 14, pp. 295–315, 1963. of homogeneous binary control vectors that result in a two- [20] N. Biggs, Algebraic Graph Theory. Cambridge University Press, 1974. [21] R. Diestel, Graph Theory. Springer-Verlag New York, Inc., 1997. dimensional controllable subspace. Finally, we gave a lower- [22] C. Godsil and R. Gordon, Algebraic Graph Theory. Springer, 2001. bound on the dimension of the controllable subspace in terms [23] A. Lubiw, “Some np-complete problems similar to graph isomorphism,” of the distance of the followers to the leaders. SIAM Journal on Computing, vol. 10, no. 1, pp. 11–21, 1982. [24] C. Colbourn and A. Rosa, Triple Systems. Clarendon Press Oxford, There are several natural open problems we plan to in- 1999. vestigate further. The characterization of the graphs that are [25] T. Kirkman, “On a problem in combinatorics,” Cambridge Dublin Math. completely uncontrollable and have distinct eigenvalues is a J., vol. 2, pp. 191–204, 1847. [26] R. Wilson, “Nonisomorphic steiner triple systems,” Mathematische very interesting problem. From a design perspective, the class Zeitschrift, vol. 135, pp. 303–313, 1974. of essentially controllable graphs are robust to the choice of [27] L. Babai, “Almost all Steiner triple systems are asymmetric,” Annals of input vertices; hence finding sufficient conditions for an asym- Discrete Mathematics, vol. 7, pp. 37–39, 1980. [28] C. Linder and A. Rosa, “On the existence of automorphism free steiner metric graph to be essentially controllable and investigating triple systems,” Journal of Algebra, vol. 34, pp. 430–443, 1975. if the class of essentially controllable graphs asymptotically [29] R. Fisher, “An examination of the different possible solutions of a approaches the class of asymmetric graphs are of great impor- problem in incomplete blocks,” Annals of Human Genetics, vol. 10, pp. 52–75, 1940. tance. Investigating the existence of a polynomial-time algo- [30] L. Babai, “Combinatorial optimization,” in Handbook of Combinatorics, rithm for generating essentially controllable graphs, exploring Volume II, Chapter 27 (R. L. Graham, M. Gr¨otschel, and L. Lov´asz, scenarios with multiple leaders, and extending the proposed eds.), Elsevier (North-Holland), 1995. [31] N. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences. classifications to other, possibly nonlinear, networked control Academic Press, 1995. systems are other areas of future work. APPENDIX REFERENCES A. Proof of Proposition 3.1 [1] C. Aguilar and B. Gharesifard, “A graph-theoretic classification for the controllability of the laplacian leader-follower dynamics,” in IEEE Conf. Here, we provide an alternative proof of Proposition 3.1. on Decision and Control, (Los Angles, CA), 2014. submitted. The proof is an easy consequence of the following result. [2] M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Lemma A.1: (Invariance of graph automorphisms under the Networks. Applied Mathematics Series, Princeton University Press, 2010. Laplacian matrix): If P is an automorphism of G and Pv = v [3] W. Ren and Y. Cao, Distributed Coordination of Multi-Agent Networks. for some v ∈ Rn then PLv = Lv. In other words, Communications and Control Engineering, New York: Springer, 2011. the subspace of fixed elements of P is invariant under the [4] F. Bullo, J. Cort´es, and S. Mart´ınez, Distributed Control of Robotic L Networks. Applied Mathematics Series, Princeton University Press, Laplacian matrix . 2009. Electronically available at http://coordinationbook.info. Proof: If P is an automorphism of G then P commutes [5] M. O. Jackson, Social and Economic Networks. Princeton University with both the adjacency matrix A and the degree matrix D, Press, 2010. L [6] C.-T. Lin, “Structural controllability,” IEEE Transactions on Automatic and therefore it also commutes with the Laplacian matrix . Control, vol. 19, no. 3, pp. 201–208, 1974. Therefore, PLv = LPv = Lv. [7] H. Tanner, “On the controllability of nearest neighbor interconnections,” We now prove Proposition 3.1. in IEEE Conf. on Decision and Control, pp. 2467–2472, 2004. b [8] A. Rahmani, M. Ji, M. Mesbahi, and M. Egerstedt, “Controllability of Proof of Proposition 3.1: If is leader symmetric then multi-agent systems from a graph-theoretic perspective,” SIAM, vol. 48, there exists an automorphism P such that Pb = b. Applying no. 1, pp. 162–186, 2009. Lemma A.1 recursively we have that PLkb = Lkb for any [9] Z. Ji, Z. Wang, H. Lin, and Z. Wang, “Interconnection topologies for N L b multi-agent coordination under leader-follower framework,” Automatica, k ∈ 0. Thus, the subspace h ; i lies in the eigenspace of vol. 45, pp. 2857–2863, 2009. P associated to the eigenvalue λ = 1. Because P is not the identity matrix it follows that hL; bi is a strict subspace of Case 1. Suppose that the number of distinct indices where n R . b1, b2, b3 are nonzero is three. Say b1 is nonzero at the pair of indices (i, j), b2 is nonzero at the pair of indices (j, k), B. Proof of Proposition 4.3 and thus b3 is nonzero at the pair of indices (i, k). Then from u ⊥{b , b , b } we obtain Case n = 4: Let L be the Laplacian matrix of a 2 1 2 3 4 connected graph that is completely uncontrollable on {0, 1} . 1 1 0 u (i) 0 L 2 Suppose by contradiction that has distinct eigenvalues 0 1 1 u2(j) = 0 , R4 0 = λ1, λ2, λ3, λ4. Then any basis of of eigenvectors 1 0 1 u (k) 0 L 2 of is uniquely determined up to scalar multiples. Let 4 u u u {u1, u2, u3, u4} be basis of R consisting of mutually orthog- whose unique solution is 2(i) = 2(j) = 2(k) = 0. onal eigenvectors of L. Then by Proposition 2.1(ii), the basis Therefore, since u2 ⊥ u1 we have (possibly after permuting 4 u u {u1, u2, u3, u4} annihilates {0, 1} . Without loss of generality indices) that 2 = [1 − 1 0 0 0]. Now, 2 is orthogonal to T b T b T b T (w.l.o.g.), let u1 = [1, 1, 1, 1] . Then {u2, u3, u4} annihi- 1 = [0 0 1 1 0] , 2 = [0 0 1 0 1] , 3 = [0 0 0 1 1] , and 4 b T lates the standard basis vectors {e1, e2, e3, e4}⊂ {0, 1} . also 4 = [1 1 0 0 0] . This leaves the following six vectors Therefore, by the pigeon hole principle, one of the vectors in S2 that are annihilated by {u3, u4, u5}: u u u u in { 2, 3, 4}, say 2, must contain two zero entries. Then, T T T b5 = [1 0 1 0 0] , b6 = [1 0 0 1 0] , b7 = [1 0 0 0 1] , since u2 ⊥ u1, we have (possibly after permuting coordinates) T b T b T b T that u2 = [1, −1, 0, 0] . Without loss of generality, we may 8 = [0 1 1 0 0] , 9 = [0 1 0 1 0] , 10 = [0 1 0 0 1] . now assume that u ⊥ e , and since u ⊥ {u , u }, it 3 1 3 1 2 Now, we may assume that u ⊥ e , and since also u ⊥ follows that u is of the form u = [0, 0, 1, −1]T . Finally, 3 1 3 3 3 {u , u }, then u must be of the form u = [0 0 c c c ]T since u ⊥ {u , u , u }, then u = [1, 1, −1, −1]T . Now 1 2 3 3 1 2 3 4 1 2 3 4 where c + c + c = 0. Now, by the pigeon hole principle, put, 1 2 3 u3 is orthogonal to at least two vectors in {b5,..., b10}. We U 1 u 1 u 1 u 1 u claim that in fact u is orthogonal to exactly two of them. = u1 1 u2 2 u3 3 u4 4 . 3 k k k k k k k k Indeed, suppose that u3 ⊥ b5. Then clearly c1 = 0 and thus Lh U UT i Then, since = diag([0, λ2, λ3, λ4]) , we obtain that c2 = −c3 6=0. Now, in this case we also have that u3 ⊥ b8. u b u b X X If 3 ⊥ 6 or 3 ⊥ 7 then clearly c2 = c3 = 0, which 1 2 u 0 u b L = , where is a contradiction since 3 6= . Similarly, if 3 ⊥ 9 or XT X u b " 2 3# 3 ⊥ 10 then again c2 = c3 = 0, which is a contradiction. Thus, if u3 ⊥ b5 then u3 ⊥ b8, and u3 is not orthogonal to 1 1 1 {b6, b7, b8, b9, b10}. Similar arguments show that if u3 ⊥ b6 λ3 + 4 λ4 − 2 λ3 + 4 λ4 X1 = , 1 1 1 1 then u3 ⊥ b9, and u3 is not orthogonal to any vector in − 2 λ3 + 4 λ4 2 λ3 + 4 λ4 {b6, b7, b8, b10}, and that if u3 ⊥ b7 then u3 ⊥ b10, λ4 1 1 X = − , and u is not orthogonal to any vector in {b , b , b , b }. 2 4 1 1 3 5 6 8 9 Hence, after a possible permutation of the indices, we have 1 λ + 1 λ − 1 λ + 1 λ u T u b b X = 2 2 4 4 2 2 4 4 . that 3 = [0, 0, 0, 1, −1] , and therefore 3 ⊥ { 5, 8}. 3 − 1 λ + 1 λ 1 λ + 1 λ u u u u u 2 2 4 4 2 2 4 4 Now, since 4 ⊥ { 1, 2, 3}, it follows that 4 takes the u u Since λ4 > 0, and the off diagonal entries of L are either form 4 = (a, a, −2(a + b),b,b) for a,b 6= 0. Now, 4 b b b b 0 or −1, we obtain from the (1, 3) entry of L that λ4 = 4. is orthogonal to one of 6, 7, 9, 10. It is readily verified Then, from the (1, 2) entry of L, either λ3 = 2 or λ3 = 4. that in any case this implies that a = −b. Therefore, we u T In the latter case, L has a repeated eigenvalue, which is a have that 4 = [1 1 0 − 1 − 1] , and this implies that u T contradiction, and therefore λ3 = 2. Then, from the (3, 4) 5 = [1 1 − 4 1 1] . Now put entry of L, the only possible cases are λ = 2 or λ = 4. 2 2 U 1 u 1 u 1 u 1 u 1 u L = u1 1 u2 2 u3 3 u4 4 u5 5 In either case, has a repeated eigenvalue, which leads to a k k k k k k k k k k contradiction. This completes the proof for n =4. h i L U UT Case : Let L be the Laplacian matrix of a and compute = diag([0, λ2, λ3, λ4, λ5]) . The first n = 5 L connected graph that is completely uncontrollable on {0, 1}5. column of is Suppose by contradiction that L has distinct eigenvalues 1 1 1 2 λ2 + 4 λ4 + 20 λ5 . Then, by Proposition 2.1(ii), there 1 1 1 0 = λ1, λ2, λ3, λ4, λ5 − 2 λ2 + 4 λ4 + 20 λ5 exists an orthogonal set {u , u , u , u , u } of eigenvectors L 1 1 2 3 4 5 1 = − 5 λ5 . of L that annihilates {0, 1}5. Without loss of generality, let 1 1 − 4 λ4 + 20 λ5 u = [1 1 1 1 1]T . Then {u , u , u , u } annihilates the set 1 1 1 2 3 4 5 − 4 λ4 + 20 λ5 S = {b ∈ {0, 1}5 | kbk = 2}, i.e., the set of elements in 2 1 L {0, 1}5 containing two nonzero entries. Now |S | = 5 = 10, From the third component of 1, we see that because 0 < λ5 2 2 L and therefore by the pigeon hole principle, there is at least and the off diagonal entries of are either 0 or -1, we must have λ5 =5. The updated L is one vector in {u2, u3, u4, u5}, say u2, that is orthogonal to at least three vectors in S2. Hence, let b1, b2, b3 ∈ S2 be X X u b b b 1 2 distinct vectors such that 2 ⊥{ 1, 2, 3}. There are three L = , XT X cases to consider. " 2 3# where Now, by the pigeon hole principle, u3 is orthogonal to at b b 1 λ + 1 λ + 1 − 1 λ + 1 λ + 1 −1 least two vectors in { 5,..., 10}. Similar arguments as in 2 2 4 4 4 2 2 4 4 4 u X = − 1 λ + 1 λ + 1 1 λ + 1 λ + 1 −1 , Case 1. show that in fact 3 is orthogonal to four vectors 1 2 2 4 4 4 2 2 4 4 4 b b u T −1 −1 4 in { 5,..., 10} and that 3 = [−1 1 0 0 0] . Now, we may assume that u4 ⊥ e1, and since also u4 ⊥{u1, u2, u3}, 1 1 1 u T X2 = (1 − λ4) , this implies that 4 = [0 0 − 1 1 0] . This then fixes 4 1 1 T u5 = [−1, −1, −1, −1, 4] . The proof that L has a repeated 1 1 1 1 1 1 X 2 λ3 + 4 λ4 + 4 − 2 λ3 + 4 λ4 + 4 eigenvalue is similar as in Case 1. 3 = 1 1 1 1 1 1 . − λ3 + λ4 + λ3 + λ4 + Case 3. Suppose that the number of distinct indices where 2 4 4 2 4 4 b , b , b are nonzero is 5, say at (i, j), (k,ℓ), (m, k), respec- Consider the (1, 5) entry of L, namely − 1 λ + 1 . The only 1 2 3 4 4 4 tively. Then from u ⊥ {b , b , b } we obtain the linear possible cases are that λ = 5 or λ = 1. In the case λ = 2 1 2 3 4 4 4 system 5 we have a repeated eigenvalue, which is a contradiction. u Therefore, λ4 =1, and the updated Laplacian matrix is 2(i) 0 u X X 11000 2(j) 0 1 2 u L 00110 2(k) = 0 , = , where u "XT X # 00101 2(ℓ) 0 2 3 u 2(m) 0 1 1 1 1 whose solution space is 2-dimensional and spanned by the 2 λ2 + 2 − 2 λ2 + 2 −1 1 1 1 1 v T v T X1 = − λ2 + λ2 + −1 , vectors 1 = [−1 1 0 0 0] and 2 = [0 0 − 1 1 1] . 2 2 2 2 u u −1 −1 4 Since the entries of 2 sum up to one, we have that 2 = [−1 1 0 0 0]T . The rest of the proof is then identical as in X2 = 02 2, × Case 1. This completes the proof of n =5. 1 λ + 1 − 1 λ + 1 X = 2 3 2 2 3 2 . 3 − 1 λ + 1 1 λ + 1 2 3 2 2 3 2 C. Proof of Proposition 4.3 L 1 1 N Now consider the (4, 5) entry of , namely − 2 λ3 + 2 . The The proof is by induction on r ∈ . Let i, j ∈ V and only possible cases are λ3 =1 or λ3 =3. In the former case suppose that 1 ≤ d (i, j). Then necessarily i 6= j and thus 0 G we have a repeated eigenvalue, which is a contradiction, and (L )ij =0 and (L)ij = (D)ij −(A)ij = −(A)ij . This proves therefore λ3 =3. The updated Laplacian is the claim for r =1. Suppose by induction that the claim holds for r ≥ 1. Fix vertices i, j with r +1 ≤ d (i, j). Clearly, r ≤ 1 λ + 1 − 1 λ + 1 −1 0 0 2 2 2 2 2 2 d (i, j) and therefore, by the induction hypothesis,G (Lk) =0 − 1 λ + 1 1 λ + 1 −1 0 0 ij 2 2 2 2 2 2 ifG0 ≤ k < r, and in particular, (Lr 1) =0. Let N denote L = −1 −1 4 −1 −1 . − ij i the neighbors of i. Then, for any k ≥ 1 we have 0 0 −1 2 −1 k T k T k 1 T k 1 0 0 −1 −1 2 (L ) = e L e = e DL − e − e AL − e , ij i j i j i j T k 1 T k 1 L 1 1 = d e (L − )e − e L − e , Finally, consider the (1, 2) entry of , namely − 2 λ2 + 2 . The i i j ℓ j only two possible cases are λ =1 or λ =3. In the former ℓ i 2 2 X∈N case, we have the repeated eigenvalue and in k 1 k 1 λ2 = λ4 = 1 = di(L − )ij − (L − )ℓj . the latter case we have the repeated eigenvalue . λ2 = λ3 =3 ℓ i In either case, we obtain a contradiction. This completes the X∈N Now, for each ℓ ∈ N it is clear that r ≤ d (ℓ, j). Hence, proof of Case 1. i we can apply the induction hypothesis forG each ℓ ∈ N , Case 2. Suppose that the number of distinct indices where i and in particular, (Lr 1) = 0 and (Lr) = (−1)r(Ar) . b , b , b are nonzero is 4, say at (i, j), (k,ℓ), (i, k), respec- − ℓj ℓj ℓj 1 2 3 Therefore, tively. Then from u2 ⊥ {b1, b2, b3} we obtain the linear r r 1 r 1 system (L )ij = di(L − )ij − (L − )ℓj =0
ℓ i u2(i) 0 X∈N 1 10 0 and consequently u2(j) 0 0 01 1 u = , 2(k) 0 Lr+1 Lr Lr Lr 1 01 0 ( )ij = di( )ij − ( )ℓj = − ( )ℓj . u2(ℓ) 0 ℓ i ℓ i X∈N X∈N whose solution up to a scalar is u2(i) = u2(ℓ)=1 and Hence u u u u 2(j) = 2(k) = −1. Since 2 ⊥ 1, then after a possible Lr+1 r Ar r+1 eT Are T ( )ij = − (−1) ( )ℓj = (−1) ℓ j, permutation of the indices, we have that u2 = [1 1 −1 −1 0] . ℓ i ℓ i Now, u is orthogonal to b = [1 0 1 0 0]T , b = ∈N ∈N 2 1 2 X r+1 XT r T T T = (−1) (e A)A e , [1 0 0 1 0] , b3 = [0 1 1 0 0] , and b4 = [0 1 0 1 0] , and i j b b r+1eT Ar+1 e this leaves six vectors { 5,..., 10} in S2 that are annihilated = (−1) i ( ) j , by {u3, u4, u5}. Now, we may assume that u3 ⊥ e3, and since r+1 r+1 = (−1) (A )ij . u3 ⊥ {u1, u2}, it is straight forward to show that u3 takes T the form u3 = [a, b, 0,a + b, −2(a + b)] , where a,b 6= 0. This ends the proof.