Some spectral and quasi-spectral characterizations of distance-regular graphs

This is a new contribution to the question: Can we see from the spectrum of a graph whether it is distance-regular? By generalizing some results of Van Dam and Haemers, among others, we give some new spectral and quasi-spectral characterizations of distance-regularity. In this area of research, typical results concluding that a graph is distance regular require that G is cospectral with a distance-regular graph that satisfies certain combinatorial conditions. In contrast, we only require certain properties of the so-called preintersection numbers of G (and for some results also the average of some intersection numbers). These preintersection numbers follow from the spectrum and resemble the intersection numbers of distance-regular graphs. Among others, we show distance-regularity for graphs with large girth or large odd-girth, using the preintersection numbers. Mathematics Subject Classifications: 05E30, 05C50.