Some Spectral and Quasi-Spectral Characterizations of Distance-Regular   Graphs
  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. 
A. Abiad, E.R. van Dam, M.A. Fiol
math.CO