# Vertex-connectivity and $Q$-index of graphs with fixed girth

## TL;DR

This paper establishes optimal bounds on the $Q$-index of graphs and their complements to determine various connectivity properties, including $k$-connectivity, maximal connectivity, and super-connectivity, especially in triangle-free graphs.

## Contribution

It provides the first sharp bounds on the $Q$-index for connected graphs with fixed girth to ensure specific connectivity levels, extending spectral graph theory results.

## Key findings

- Derived best possible bounds for $q(G)$ and $q(ar{G})$ related to $k$-connectivity.
- Established bounds for $q(G)$ and $q(ar{G})$ to guarantee maximal and super-connectivity.
- Analyzed bounds for triangle-free graphs to ensure various connectivity properties.

## Abstract

Let $q(G)$ denote the $Q$-index of a graph $G$, which is the largest signless Laplacian eigenvalue of $G$. We prove best possible upper bounds of $q(G)$ and best possible lower bounds of $q(\overline{G})$ for a connected graph $G$ to be $k$-connected and maximally connected, respectively. Similar upper bounds of $q(G)$ and lower bounds of $q(\overline{G})$ to assure $G$ to be super-connected are also obtained. Upper bounds of $q(G)$ and lower bounds of $q(\overline{G})$ to assure a connected triangle-free graph $G$ to be $k$-connected, maximally connected and super-connected are also respectively investigated.