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.