Discrete Applied Mathematics 229 (2017) 46–54 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Addressing graph products and distance-regular graphs Sebastian M. Cioabă a, Randall J. Elzinga b,1 , Michelle Markiewitz a, Kevin Vander Meulen c,*, Trevor Vanderwoerd c,2 a Department of Mathematical Sciences, University of Delaware, Newark, DE 19716-2553, USA b Department of Mathematics, Royal Military College, Kingston, ON K7K 7B4, Canada c Department of Mathematics, Redeemer University College, Ancaster, ON L9K 1J4, Canada article info a b s t r a c t Article history: Graham and Pollak showed that the vertices of any connected graph G can be assigned Received 20 September 2016 t-tuples with entries in f0; a; bg, called addresses, such that the distance in G between Received in revised form 11 May 2017 any two vertices equals the number of positions in their addresses where one of the Accepted 31 May 2017 addresses equals a and the other equals b. In this paper, we are interested in determining Available online 1 July 2017 the minimum value of such t for various families of graphs. We develop two ways to obtain To the memory of David A. Gregory this value for the Hamming graphs and present a lower bound for the triangular graphs. ' 2017 Elsevier B.V. All rights reserved. Keywords: Distance matrix Spectrum Triangular graphs Hamming graphs Graph addressing 1. Graph addressings A t-address is a t-tuple with entries in f0; a; bg. An addressing of length t for a graph G is an assignment of t-addresses to the vertices of G so that the distance between two vertices is equal to the number of locations in the addresses at which one of the addresses equals a and the other address equals b. For example, we have a 3-addressing of a graph in Fig. 1. Graham and Pollak [13] introduced such addressings, using symbols {∗; 0; 1g instead of f0; a; bg, in the context of loop switching networks. We are interested in the minimum t such that G has an addressing of length t. We denote such a minimum by N(G). Graham and Pollak [13,14] showed that N(G) equals the biclique partition number of the distance multigraph of G. Specifically, the distance multigraph of G, D(G), is the multigraph with the same vertex set as G where the multiplicity of any edge uv equals the distance between vertices u and v in G. The biclique partition number bp(H) of a multigraph H is the minimum number of complete bipartite subgraphs (bicliques) of H whose edges partition the edge set of H. This parameter and its covering variations have been studied by several researchers and appear in different contexts such as computational complexity or geometry (see for example, [8,13–16,19,21,22,25]). Graham and Pollak deduced that N(G) ≤ r(n − 1) for any connected graph G of order n and diameter r and conjectured that N(G) ≤ n − 1 for any connected graph G of order n. This conjecture, also known as the squashed cube conjecture, was proved by Winkler [24]. * Corresponding author. E-mail addresses: [email protected] (S.M. Cioabă), [email protected] (R.J. Elzinga), [email protected] (M. Markiewitz), [email protected] (K. Vander Meulen), [email protected] (T. Vanderwoerd). 1 Current address: Info-Tech Research Group, London, ON, N6B 1Y8, Canada. 2 Current address: Department of Civil and Environmental Engineering, University of Waterloo, ON N2L 3G1, Canada. http://dx.doi.org/10.1016/j.dam.2017.05.018 0166-218X/' 2017 Elsevier B.V. All rights reserved. S.M. Cioabă et al. / Discrete Applied Mathematics 229 (2017) 46–54 47 Fig. 1. A graph addressing of length 3. To bound N(G) below, Graham and Pollak used an eigenvalue argument on the adjacency matrix of D(G). Specifically, if M is a symmetric real matrix, let nC(M); n−(M), and n0(M) denote the number of eigenvalues of M (including multiplicity) that are positive, negative and zero, respectively. The inertia of M is the triple (nC(M); n0(M); n−(M)). The adjacency matrix of D(G) will be denoted by D(G); we will also refer to D(G) as the distance matrix of G. The inertia of distance matrices has been studied by various authors for many classes of graphs [3,17,18,26]. Witsenhausen (cf. [13,14]) showed that N(G) ≥ maxfnC(D(G)); n−(D(G))g: (1) Letting Jn denote the all one n × n matrix and In denote the n × n identity matrix, and observing that n−(D(Kn)) D n−(Jn − In), Graham and Pollak [13,14] used the bound (1) to conclude that N(Kn) D n − 1: (2) Graham and Pollak [13,14] also determined N(Kn;m) for many values of n and m. The determination of N(Kn;m) for all values of n and m was completed by Fujii and Sawa [11]. A more general addressing scheme, allowing the addresses to contain more than two different nonzero symbols, was recently studied by Watanabe, Ishii and Sawa [23]. The parameter N(G) has been determined when G is a tree or a cycle [14], as well as one particular triangular graph T4 [25], described in Section5. For the Petersen graph P, Elzinga, Gregory and Vander Meulen [10] showed that N(P) D 6. To the best of our knowledge, these are the only graphs G for which addressings of length N(G) have been determined. We will say a t-addressing of G is optimal if t D N(G). An addressing is eigensharp [19] if equality is achieved in (1). In this paper, we study optimal addressings of Cartesian graph products and the distance-regular graphs known as triangular graphs. Let H(n; q) be the Hamming graph whose vertices are the n-tuples over an alphabet with q letters with two n-tuples being adjacent if and only if their Hamming distance is 1. We give two different proofs showing that N(H(n; q)) D n(q − 1). This generalizes the Graham–Pollak result (2) since H(1; q) D Kq. We show that the triangular graphs are not eigensharp. 2. Addressing Cartesian products Let G D G1□G2□ ··· □Gk denote the Cartesian product of graphs G1; G2;:::; Gk. Then G has vertex set V (G) D f(v1; v2; : : : ; vk) j vi 2 V (Gi)g. Two vertices v D (v1; : : : ; vk) and u D (u1;:::; uk) of G are adjacent if for some index j, vj is adjacent to uj in Gj while vi D ui for all remaining indices i 6D j. Thus, if d and di denote distances between pairs of vertices in G and Gi respectively, then for every v; u 2 V (G), k X d(v; u) D di(vi; ui): (3) iD1 It follows that if each Gi, i D 1;:::; k, is given an addressing, then each vertex x of G may be addressed by concatenating the addresses of its components xi. Therefore, the parameter N is sub-additive on Cartesian products; that is, if G D G1□ ··· □Gk (4) then N(G) ≤ N(G1) C···C N(Gk): (5) ( ) ( ) C···C ≤ Pk − ≤ Qk − D − Note that N(G1) N(Gk) iD1ni k iD1ni 1 n 1. Thus (5) can improve on Winkler's upper bound of n − 1 when G is a Cartesian product. Question 2.1. Must equality hold in (5) for all choices of Gi? Remark 3.4 might provide a possible counterexample. 3. Distance matrices of cartesian products In this section we determine N(G) when G is the Cartesian product of complete graphs. We first develop some results about the inertia of the distance matrix of a Cartesian product.3 3 The approach we take is due to the late D.A. Gregory. 48 S.M. Cioabă et al. / Discrete Applied Mathematics 229 (2017) 46–54 If v1; : : : ; vn denote the vertices of a connected graph G, the distance matrix D(G) of G is the n × n matrix with entries D(G)ij D d(vi; vj). Because G is connected, its adjacency matrix A(G) and its distance matrix D(G) are irreducible symmetric nonnegative integer matrices and by the Perron–Frobenius Theorem (see [5, Proposition 3.1.1] or [12, Theorem 8.8.1]), the largest eigenvalue of each of these matrices has multiplicity 1. We call this largest eigenvalue the Perron value of the matrix and often denote it by ρ. To obtain a formula for the distance matrix of a Cartesian product of graphs, we will use an additive analogue of the Kronecker product of matrices. Note that if A is an n × m matrix and c 2 R, then c C A is the n × m matrix cJ C A with J n m the all one n × m matrix. Further, recall that if A is an n × n matrix and B an m × m matrix, with x 2 R , y 2 R , then the Kronecker products A ⊗ B and x ⊗ y are defined as 2a11B a12B ··· a1mB3 2x1y3 ··· 6a21B a22B a2mB7 6x2y7 A ⊗ B D 6 : : : 7 and x ⊗ y D 6 : 7 : (6) 4 : : : 5 4 : 5 am1B am2B ··· ammB xny For the additive analogue, we use the symbol ⋄ and define A ⋄ B and x ⋄ y as 2a11 C B a12 C B ··· a1m C B3 2x1 C y3 C C ··· C C 6a21 B a22 B a2m B7 6x2 y7 A ⋄ B D 6 : : : 7 and x ⋄ y D 6 : 7 : (7) 4 : : : 5 4 : 5 am1 C B am2 C B ··· amm C B xn C y If G D G1□G2□ ··· □Gk, then the additive property (3) implies that D(G) D D(G1) ⋄ D(G2) ⋄ · · · ⋄ D(Gk): (8) Note that G1□G2 is isomorphic to G2□G1 and, equivalently, D(G1) ⋄ D(G2) is permutationally similar to D(G2) ⋄ D(G1).