The Geometry of t-Cliques in k-Walk-Regular Graphs

A graph is walk-regular if the number of cycles of length l rooted at a given vertex is a constant through all the vertices. For a walk-regular graph G with d+1 different eigenvalues and spectrally maximum diameter D = d, we study the geometry of its d-cliques, that is, the sets of vertices which are mutually at distance d. When these vertices are projected onto an eigenspace of its adjacency matrix, we show that they form a regular tetrahedron and we compute its parameters. Moreover, the results are generalized to the case of k-walk-regular graphs, a family which includes both walkregular and distance-regular graphs, and their t-cliques or vertices at distance t from each other. 1 Preliminaries Distance-regular graphs with diameter D can be characterized by the invariance of the number of walks of length l ≥ t between vertices at a given distance t, 0 ≤ t ≤ D (see e.g. Rowlinson [13] or Fiol[5]). Similarly, walk-regular graphs are characterized by the fact that the number of closed walks of length l ≥ 0 rooted at any given vertex u does not depend on u. Thus, a distance-regular graph is also walk-regular, but the converse, in general, is not true (see e.g. Godsil [10]). In this paper, we first recall some characterizations and derive some basic results on a walk-regular graph G. Afterwards, this background is used to study the geometry ∗Research supported by the Ministerio de Educación y Ciencia, Spain, and the European Regional Development Fund under projects MTM2005-08990-C02-01 and TEC2005-03575, and by the Catalan Research Council under project 2005SGR00256. 1 of the vertices which are mutually at distance d, where d + 1 is the number of different eigenvalues in the spectrum of G. More precisely, when the coordinate vectors representing such vertices of G are projected onto the eigenspace of any eigenvalue, we show that the points obtained are the vertices of a regular tetrahedron and we compute their radius (distance from the center to every vertex), edge length and angle formed by the vectors going from the center to each vertex. Then, imposing that such parameters must be nonnegative, some consequences on the eigenvalue multiplicities and the d-clique number, which is the maximum number of vertices at distance d from each other, are derived. Finally, these results are generalized for the so-called k-walk-regular graphs, which were recently introduced by the authors in [3, 7] and their t-cliques, 1 ≤ t ≤ k. These graphs are characterized by the invariance of the number of walks of length l ≥ t between vertices at a given distance t, 0 ≤ t ≤ t. Then, this family includes both walk-regular (k = 0) and distance-regular (k = D) graphs. 1.1 Background We begin with some notation and basic results. Throughout this paper,G = (V,E) denotes a simple, connected graph, with order n = |V | and adjacency matrix A. The distance between two vertices u, v is denoted by dist(u, v), so that the eccentricity of a vertex u is ecc(u) = maxv∈V dist(u, v) and the diameter of the graph is D = D(G) = maxu∈V ecc(u). The spectrum of G is spG = spA = {λ0 0 , λ m1 1 , . . . , λ md d }, where λ0 > λ1 > · · · > λd and the superscripts stand for the multiplicities mi = m(λi). In particular, note that m0 = 1 (since G is connected) and m0 +m1 + · · ·+md = n. It is well-known that the diameter of G satisfies D ≤ d (see, for instance, Biggs [1]). Then, a graph with D = d is said to have spectrally maximum diameter. For a given ordering of the vertices, the vector space of linear combinations (with real coefficients) of the vertices of G is identified with R, with canonical basis {eu : u ∈ V }. For every 0 ≤ h ≤ d, the orthogonal projection of R onto the eigenspace Eh = Ker(A − λhI) are given by the (Lagrange interpolation) polynomials of degree d Ph = 1 φh d ∏ i=0 i6 =h (x− λi) = (−1) πh d ∏ i=0 i6 =h (x− λi) (0 ≤ h ≤ d), where φh = ∏d i=0,i6 =h(λh − λi) and πh = |φh| are “moment-like” parameters satisfying d ∑ h=0 (−1) πh p(λh) = 0 (1) for any polynomial p of degree smaller than d (just observe that the coefficient of x in both terms of p(x) = ∑d h=0 p(λh)Ph(x) must be zero). In particular, recall that H = nP0 is the Hoffman polynomial, which characterizes the regularity of G by the condition H(A) = J , 2 the all-1 matrix (see Hoffman [12]). The matrices Eh = Ph(A) corresponding to these orthogonal projections onto Eh are called the (principal) idempotents of A. Then, the orthogonal decomposition of the unitary vector eu, representing vertex u, is: eu = z 0 u + z 1 u + · · · + z d u , where z h u = Ph(A)eu = Eheu ∈ Eh. From this decomposition, we define the u-local multiplicity of eigenvalue λh as mu(λh) = ‖z h u‖ 2 = 〈Eheu,Eheu〉 = 〈Eheu,eu〉 = (Eh)uu, satisfying ∑d h=0 mu(λh) = 1 and