International Journal of Mathematics and Mathematical Sciences 
Volume 23 (2000), Issue 2, Pages 89-97
doi:10.1155/S0161171200000740

The matching polynomial of a distance-regular graph

Robert A. Beezer1 and E. J. Farrell2

 1Department of Mathematics and Computer Science, University of Puget Sound, Tacoma 98416, Washington, USA
2Department of Mathematics, University of the West Indies, St. Augustine, Trinidad and Tobago
Received 15 August 1997

Abstract

A distance-regular graph of diameter d has 2d 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.