On the positive and negative inertia of weighted graphs

TL;DR

This paper investigates the positive and negative inertia indices of weighted graphs, providing methods to calculate these indices for trees, unicyclic, and bicyclic graphs, enhancing understanding of their spectral properties.

Contribution

It introduces new methods for calculating positive and negative inertia indices specifically for weighted trees, unicyclic, and bicyclic graphs.

Findings

Derived formulas for inertia indices of weighted trees

Extended calculations to unicyclic and bicyclic graphs

Enhanced spectral analysis techniques for weighted graphs

Abstract

The number of the positive, negative and zero eigenvalues in the spectrum of the (edge)-weighted graph $G$ are called positive inertia index, negative inertia index and nullity of the weighted graph $G$, and denoted by $i_+(G)$, $i_-(G)$, $i_0(G)$, respectively. In this paper, the positive and negative inertia index of weighted trees, weighted unicyclic graphs and weighted bicyclic graphs are discussed, the methods of calculating them are obtained.