On the determinant of the distance matrix of a bicyclic graph

TL;DR

This paper derives a formula for the determinant of the distance matrix of a bicyclic graph with disjoint cycles, extending previous results for trees and unicyclic graphs.

Contribution

It provides a new explicit formula for the determinant of the distance matrix in a specific class of bicyclic graphs with disjoint cycles.

Findings

Derived a formula for the determinant of the distance matrix of disjoint-cyclic bicyclic graphs.

Extended known formulas from trees and unicyclic graphs to a broader class.

Contributed to the theoretical understanding of graph distance matrices.

Abstract

Two cycles are referred as disjoint if they have no common edges. In this paper, we will investigate the determinant of the distance matrix of a graph, giving a formula for the determinant of the distance matrix of a bicyclic graph whose two cycles are disjoint, which extends the formula for the determinant of the distance matrix of a tree, as well as that of a unicyclic graph.