Linear time algorithm to check the singularity of block graphs

TL;DR

This paper presents a linear time algorithm to determine the singularity of block graphs, a class of graphs where each block is a complete graph, addressing an open problem in graph theory.

Contribution

The paper introduces the first linear time algorithm for checking the singularity of block graphs, solving a problem posed in 2013.

Findings

Algorithm runs in linear time

Efficiently characterizes singularity of block graphs

Addresses an open problem in graph theory

Abstract

A block graph is a graph in which every block is a complete graph. Let $G$ be a block graph and let $A(G)$ be its (0,1)-adjacency matrix. Graph $G$ is called nonsingular (singular) if $A(G)$ is nonsingular (singular). Characterizing nonsingular block graphs is an interesting open problem proposed by Bapat and Roy in 2013. In this article, we give a linear time algorithm to check whether a given block graph is singular or not.