Connected signed graphs with given inertia indices and given girth

TL;DR

This paper investigates the relationship between girth and negative inertia index in connected signed graphs, establishing bounds, characterizing extremal graphs, and determining graphs with specified inertia indices and girth.

Contribution

It provides new bounds on the negative inertia index based on girth and characterizes extremal graphs achieving these bounds, also determining graphs with given inertia indices and girth.

Findings

Proves that the negative inertia index is at least half the girth minus one.

Characterizes extremal signed graphs that attain the lower bound of the negative inertia index.

Determines connected signed graphs with specified positive inertia index, nullity, and girth.

Abstract

Suppose that $\Gamma=(G, \sigma)$ is a connected signed graph with at least one cycle. The number of positive, negative and zero eigenvalues of the adjacency matrix of $\Gamma$ are called positive inertia index, negative inertia index and nullity of $\Gamma$, which are denoted by $i_+(\Gamma)$, $i_-(\Gamma)$ and $\eta(\Gamma)$, respectively. Denoted by $g$ the girth, which is the length of the shortest cycle of $\Gamma$. We study relationships between the girth and the negative inertia index of $\Gamma$ in this article. We prove $i_{-}(\Gamma)\geq \lceil\frac{g}{2}\rceil-1$ and extremal signed graphs corresponding to the lower bound are characterized. Furthermore, the signed graph $\Gamma$ with $i_{-}(\Gamma)=\lceil\frac{g}{2}\rceil$ for $g\geq 4$ are given. As a by-product, the connected signed graphs with given positive inertia index, nullity and given girth are also determined, respectively.