A q-analogue of Graham, Hoffman and Hosoya's Theorem

Abstract

Graham, Hoffman and Hosoya gave a very nice formula about the determinant of the distance matrix D(G) of a graph G in terms of the distance matrix of its blocks. We generalize this result to a q-analogue of D(G). Our generalization yields results about the equality of the determinant of the mod-2 (and in general mod-k) distance matrix (i.e. each entry of the distance matrix is taken modulo 2 or k) of some graphs. The mod-2 case can be interpreted as a determinant equality result for the adjacency matrix of some graphs.