Computing the Determinant of the Distance Matrix of a Bicyclic Graph

摘要

LetGbe a connected graph with vertex setV= {v1, ...,vn}. The distanced(vi,vj) between two verticesviandvjis the number of edges of a shortest path linking them. The distance matrix ofGis then×nmatrix such that its (i,j)-entry is equal tod(vi,vj). A formula to compute the determinant of this matrix in terms of the number of vertices was found when the graph either is a tree or is a unicyclic graph. For a byciclic graph, the determinant is known in the case where the cycles have no common edges. In this paper, we present some advances for the remaining cases; i.e., when the cycles share at least one edge. We also present a conjecture for the unsolved cases.