Distance mean-regular graphs

We introduce the concept of distance mean-regular graph, which can be seen as a generalization of both vertex-transitive and distance-regular graphs. A graph Γ = (V,E) with diameter D is distance meanregular when, for given u ∈ V , the averages of the intersection numbers ai(u, v), bi(u, v), and ci(u, v) (defined as usual), computed over all vertices v at distance i = 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, but the converse does not hold, and give a condition for the converse to be 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. Moreover, these matrices are computed from a sequence of orthogonal polynomials evaluated at A, the adjacency matrix of Γ.