Equivalent Characterizations of the Spectra of Graphs and Applications to Measures of Distance-regularity

Abstract

The spectrum of a graph usually provides a lot of information about its combinatorial structure. Moreover, from the spectrum, the so-called predistance polynomials can be defined, as a generalization, for any graph, of the distance polynomials of a distance-regular graph. Going further, the preintersection numbers generalize the intersection numbers of a distance-regular graph. This paper describes, for any graph, the closed relationships between its spectrum, predistance polynomials, and preintersection numbers. Then, some applications to derive combinatorial properties of the given graph, most of them related to some fundamental characterizations of distance-regularity, are presented. In particular, the so-called `spectral excess theorem' is revisited. This result states that a connected regular graph is distance-regular if and only if its spectral excess, which is a value computed from the spectrum, equals the average excess, that is, the mean of the numbers of vertices at maximum distance from every vertex.