The matching polynomial of a distance‐regular graph

Abstract

A distance‐regular graph of diameter d has 2 d intersection numbers that determine many properties of graph (e.g., its spectrum). We show that the first six coefficients of the matching polynomial of a distance‐regular graph can also be determined from its intersection array, and that this is the maximum number of coefficients so determined. Also, the converse is true for distance‐regular graphs of small diameter—that is, the intersection array of a distance‐regular graph of diameter 3 or less can be determined from the matching polynomial of the graph.