Distance mean-regular graphs - Diego, V.; Fiol, M. A. (Designs, codes, and cryptography, 2017)
We introduce the concept of distance mean-regular graph, which can be seen as a generalization of both vertex-transitive and distance-regular graphs. Let Γ be a graph with vertex set V , diameter D , adjacency matrix A , and adjacency algebra A . Then, Γ is distance mean-regular when, for a given u ∈ V , the averages of the intersection numbers p i j h ( u , v ) = | Γ i ( u ) ∩ Γ j ( v ) | (number of vertices at distance i from u and distance j from v ) computed over all vertices v at a given distance h ∈ { 0 , 1 , … , D } from u , do not depend on u . In this work we study some properties and characterizations of these graphs. For instance, it is shown that a distance mean-regular graph is always distance degree-regular, and we give a condition for the converse to be also true. Some algebraic and spectral properties of distance mean-regular graphs are also investigated. We show that, for distance mean regular-graphs, the role of the distance matrices of distance-regular graphs is played for the so-called distance mean-regular matrices. These matrices are computed from a sequence of orthogonal polynomials evaluated at the adjacency matrix of Γ and, hence, they generate a subalgebra of A . Some other algebras associated to distance mean-regular graphs are also characterized.