View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector DISCRETE APPLIED MATHEMATICS ELSEVIEK Discrete Applied Mathematics 65 (1996) 2555273 Laplacian spectra and spanning trees of threshold graphs P.L. Hammer*, A.K. Kelmans RUTCOR, Rutgers Uniwrsit). NW Bruns~t~clr. NJ OXY03, L’S.4 Received 12 May 1993; revised 4 March 1994 Abstract In this paper we study the Laplacian spectra, the Laplacian polynomials, and the number of spanning trees of a special class of graphs called threshold graphs. We.give formulas for the Laplacian spectrum, the Laplacian polynomial, and the number of spanning trees of a threshold graph, in terms of so-called composition sequences of threshold graphs. It is shown that the degree sequence of a threshold graph and the sequence of eigenvalues of its Laplacian matrix are “almost the same”. On this basis, formulas are given to express the Laplacian polynomial and the number of spanning trees of a threshold graph in terms of its degree sequcncc. Moreover, threshold graphs are shown to be uniquely defined by their spectrum. and a poly- nomial time procedure is given for testing whether a given sequence of numbers is the spectrum of a threshold graph. 1. Introduction and notations The Laplacian matrix (or the matrix of admittance) of a graph is one of the classical concepts in graph theory [l, 2,5,20]. One of the most interesting and useful results about this matrix is the well-known matrix-tree theorem. which states that if 17is the order of the matrix, then the number of spanning trees of the graph is equal to the determinant of any principal minor of order n - 1. It turns out that the Laplacian polynomial of the graph, i.e. the characteristic polynomial of the Laplacian matrix of the graph (introduced in [24,25], see also [lo]), is also a very useful concept, which has a natural combinatorial interpretation. It has been shown in [30] that the inclusion-exclusion formula of [33] for the number of spanning trees of a graph can be directly “read” from this polynomial. This polynomial is also useful in evaluating the probability of connectivity of graphs. in constructing graphs having a maximum number of spanning trees. and for *Corresponding author. 0166-218X/96:$15.00 ~(‘11996-Elsevier Science W.V. All rights reserved SSDI 01 66-21 8X(94)00049-2 256 P.L. Hammer, A.K. Kelmans J Discrete Applied Mathematics 65 (1996) 255-273 various other problems in extremal graph theory [12,24426,28&30], as well as in statistical experiments (see, for example, [7]); among the papers dealing with this topic we mention [3,9,11,13,14,16,18,21~23,34437]. For the special case of regular graphs, there is a straightforward relationship between the Laplacian polynomial and the characteristic polynomial of the adjacency matrix of a graph, allowing to translate directly results concerning one of those polynomials to results concerning the other one. One of the natural questions about the Laplacian polynomial concerns the extent to which it characterizes the corresponding graph. There are many examples of cospectral graphs, i.e. non-isomorphic graphs with the same Laplacian polynomial (e.g. [12,23,24,26,28,29,34], see also [lo]). On the other hand certain classes of graphs are uniquely defined by their spectrum. Such classes include complete graphs, regular k-partite graphs, paths, circuits, trees up to 10 vertices, stars and some other special trees (e.g. [12,24,26,28-301). It is also known that in the class of regular graphs the line graphs of some combinatorial structures (complete graphs, complete bipartite graphs, block designs, projective planes, finite affine planes, etc.) are uniquely defined under certain conditions by their Laplacian polynomials among the regular graphs (e.g. [13,14,21-23,351). In this paper we study the Laplacian spectra, the Laplacian polynomials, and the number of spanning trees of a special class of graphs called threshold graphs. Thre- shold graphs were introduced in [6], and numerous properties of them were investig- ated in the literature; surveys on this topic appear in Ch. 10 of [17], in [4], and in the forthcoming monograph [32]. It is easy to see that a canonical description of a threshold graph having y1vertices can be given by a sequence of at most y1integer parameters, called its composition sequence. We give formulas for the Laplacian spectrum, the Laplacian polynomial, and the number of spanning trees of a threshold graph, in terms of its composition sequence (Theorems 4.2,4.3, and 5.4). It is shown (Theorem 5.3) that the degree sequence of a threshold graph and the sequence of eigenvalues of its Laplacian matrix are “almost the same”. On this basis, formulas are given for the Laplacian polynomial and the number of spanning trees of a threshold graph, in terms of its degree sequence (Theorem 5.4). Moreover, threshold graphs are shown to be uniquely defined by their spectrum (Theorem 6.1), and a polynomial time procedure is given for testing whether a given sequence of numbers is the spectrum of a threshold graph. All the graphs considered in this paper are undirected and have no loops or multiple edges [l, 2,201. Let V(G) and E(G) denote the set of vertices and edges of G, respectively. Let U(G) denote the number of vertices of G. 2. Decomposable graphs Given two disjoint graphs A and B, a graph G is called the sum of A and B, G = A + B, if I’(G) = V(A)uV(B) and E(G) = E(A)uE(B). Similarly, the product P.I.. Hummer. A.K. K&mans ; Discre!e Applied Muthemaiic~s 6.5 llW6) 255-273 -.157 A x B will denote the graph obtained from A + B by adding all the edges (u,h) with 0 E V(.4) and h E V(B). In particular, if B consists of a single vertex h, we write 4 + h and A x h instead of A + B and A x B: in these cases h is called an i.sol~~terland a unicwsal rertc.x, respectively. Given a graph G, let i(G) and u(G) denote the number of isolated and universal vertices, respectively. The graph G is said to be the complenrenr c?f’a grrrph G if there is a bijection ~(0:V(G) + V(G) such that (u, U) is an edge of G if and only if ((p(u), (P(U)) is not an edge of G. Let K,, be the complete graph on II vertices. and let its complement k’,, be the c’rnptl’ graph on 12vertices. If I/(K,) = I/(K,J = [g,, . ,gn) then K,, = cl1 xy? x ... xc]!,, and K,, = 8, + c/l + ... + 11,~;we shall write simply K, = .1/”and K, = MCI. A graph is called decomposable [25] if any induced subgraph having at least two v,ertices is the sum or the product of two graphs. It is easy to see that a graph is decomposable if and only if it can be obtained from one-vertex graphs by the above operations ( + ) and ( x 1, and if and only if it does not contain a path on 3 vertices as an induced subgraph. Let I;i denote the set of decomposable graphs. Clearly every disconnected graph is the (unique) sum of its connected components. Similarly. every connected decomposable graph is the (unique) product of some disconnected or one-vertex graphs (the complements of which are the connected components of the complement of the given graph). For example, the graph H in Fig. 1 is decomposable and can be represented as H = (yl + q1)(g4(q3 + q5) + qhq7). A decomposable graph G can be naturally described 1251 by a labelled rooted tree T(G) which will be called the compo,sition trcr of G. The root I’ will be labelled having the label S(T) = { x ) if G is connected. and .~(I.)= ( +) if G is disconnected. This tree can be defined recursively as follows. If G is a one-vertex graph then let T(G) be the trivial rooted tree consisting of one vertex which is the root of T(G) and has the label ( x ). Suppose that Gi is a decomposable graph, that T (Gi) is already defined for G,, and that Ti is the root of T(GJ, i = 1, . , k. Suppose also that the label .s(ri) of the root I’, of T (Gi) is (x i if Gj is connected, and is { +) if Gi is disconnected. Let T be the tree 258 P.L. Hammer, A.K. Kelmans / Discrete Applied Mathematics 65 (1996) 255-273 obtained from disjoint rooted trees T (Gi) by adding a new vertex r, the root of T, and by connecting r with every root Yi by an edge (r,rJ. If G = G1 x ... x Gk where each Gi is not the product of two graphs (which means in our case that Gi is a one-vertex or disconnected graph) then put T(G) = T, and S(V) = { x }. If G = G, + ... + Gk where each Gi is not the sum of two graphs (i.e. Gi is a connected graph) then put T(G) = T, and S(Y) = { +}. The rooted tree T(H) of the graph H in Fig. 1 is shown in Fig. 2. It is easy to see that T(G) is uniquely defined by G, and that two decomposable graphs G and F are isomorphic if and only if T(G) and T(F) are isomorphic as rooted labelled trees.