The determinant of the distance matrix of graphs with at most two cycles

TL;DR

This paper derives a formula for the determinant of the distance matrix for a broad class of graphs with up to two cycles, extending known results from trees and unicyclic graphs to more complex structures.

Contribution

It generalizes existing formulas for the distance matrix determinant to include graphs with multiple blocks, specifically those with at most two cycles.

Findings

Provides a closed-form formula for the determinant of the distance matrix for graphs with up to two cycles.

Extends previous results from trees and unicyclic graphs to more complex graphs.

Enables analysis of the distance matrix for a wider class of graphs in graph theory.

Abstract

Let $G$ be a connected graph on $n$ vertices and $D(G)$ its distance matrix. The formula for computing the determinant of this matrix in terms of the number of vertices is known when the graph is either a tree or {a} unicyclic graph. In this work we generalize these results, obtaining the determinant of the distance matrix for {all graphs} in a {class, including trees, unicyclic and bicyclic graphs. This class actually includes graphs with many cycles, provided that each block of the graph is at most bicyclic.}