On distance-regularity in graphs

Abstract If A is the adjacency matrix of a graph G , then A i is the adjacency matrix of the graph on the same vertex set in which a pair of vertices is adjacent if and only if their distance apart is i in G . If G is distance-regular, then A i is a polynomial of degree i in A . It is shown that the converse is also true. If A i is a polynomial in A , not necessarily of degree i , G is said to be distance-polynomial. It is shown that this is a larger class of graphs and some of its properties are investigated.