Some spectral and quasi-spectral characterizations of distance-regular graphs - Abiad, A.; van Dam, E.R.; Fiol, M.A. (Journal of combinatorial theory. Series A, 2016)
In this paper we consider the concept of preintersection numbers of a graph. These numbers are determined by the spectrum of the adjacency matrix of the graph, and generalize the intersection numbers of a distance-regular graph. By using the preintersection numbers we give some new spectral and quasi-spectral characterizations of distance-regularity, in particular for graphs with large girth or large odd-girth.