A \(q\)-analogue of Graham, Hoffman and Hosoya's theorem
Summary: \textit{R.L. Graham}, \textit{A.J. Hoffman}, and \textit{H. Hosoya} [``On the distance matrix of a directed graph'', J. Graph Theory 1, 85--88 (1977; Zbl 0363.05034)] 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.
scientific article; zbMATH DE number 5721430