# Nonsingular (Vertex-Weighted) Block Graphs

## TL;DR

This paper characterizes when block graphs, especially vertex-weighted ones, are nonsingular by analyzing their structure through deletion and contraction of pendant blocks, providing new insights into their algebraic properties.

## Contribution

It introduces a characterization of nonsingular vertex-weighted block graphs using reduced graphs after pendant block operations, extending previous understanding.

## Key findings

- Characterization of nonsingular vertex-weighted block graphs
- Method for analyzing nonsingularity via deletion and contraction
- Special cases for direct determination of nonsingularity

## Abstract

A graph $G$ is \emph{nonsingular (singular)} if its adjacency matrix $A(G)$ is nonsingular (singular). In this article, we consider the nonsingularity of block graphs, i.e., graphs in which every block is a clique. Extending the problem, we characterize nonsingular vertex-weighted block graphs in terms of reduced vertex-weighted graphs resulting after successive deletion and contraction of pendant blocks. Special cases where nonsingularity of block graphs may be directly determined are discussed.