LIMDA Joint Seminar: Victor Diego

Speaker: Víctor Diego, Departament de Matemàtiques, UPC

Title: Distance mean-regular graphs

Abstract:

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 G be a graph with vertex set V, diameter D, adjacency matrix A, and adjacency algebra A. Then, G is distance mean-regular when, for a given u in V, the averages of the intersection numbers p_{ij}^h(u,v)=|G_i(u)\G_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 in {0,1,ldots,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 G and, hence, they generate a subalgebra of A. Some other algebras associated to distance mean-regular graphs are also characterized.