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. Let Gamma be a graph with vertex set V, diameter D, adjacency matrix A, and adjacency algebra A. Then, Gamma is distance mean-regular when, for a given u is an element of V, the averages of the intersection numbers p(ij)(h)(u, v) = |Gamma(i)(u) boolean AND Gamma(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 is an element of{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 Gamma and, hence, they generate a subalgebra of A. Some other algebras associated to distance mean-regular graphs are also characterized.