Some classes of connected signed graphs with girth $g$ and negative inertia index $\lceil\frac{g}{2}\rceil+1$

TL;DR

This paper characterizes certain classes of connected signed graphs with a specified girth and negative inertia index, advancing understanding of their spectral properties.

Contribution

It identifies classes of connected signed graphs where the negative inertia index equals half the girth plus one, providing new spectral graph insights.

Findings

Characterization of signed graphs with negative inertia index = ⌈g/2⌉+1

Establishment of conditions relating girth and negative inertia index

Extension of spectral properties in signed graphs

Abstract

Let $\Gamma$ be a signed graph. The number of negative eigenvalues of the adjacency matrix of $\Gamma$ is called the negative inertia index of $\Gamma$, which is denoted by $i_-(\Gamma)$. The length of the shortest cycle contained in $\Gamma$ is called the girth of $\Gamma$, and it is denoted by $g$. In this paper, we give some classes of connected signed graphs $\Gamma$ which satisfy the condition $i_-(\Gamma)=\lceil\frac{g}{2}\rceil+1$.