The Q-index and connectivity of graphs

TL;DR

This paper establishes a spectral condition involving the $Q$-index that guarantees a large connected graph with fixed minimum degree is $k$-connected, extending classical Dirac-type connectivity results.

Contribution

It provides a tight sufficient spectral condition based on the $Q$-index for $k$-connectivity in large graphs with fixed minimum degree, a novel spectral approach.

Findings

Provides a tight sufficient condition for $k$-connectivity

Extends Dirac-type connectivity conditions to spectral domain

Applicable to large graphs with fixed minimum degree

Abstract

A connected graph $G$ is said to be $k$-connected if it has more than $k$ vertices and remains connected whenever fewer than $k$ vertices are deleted. In this paper, for a connected graph $G$ with sufficiently large order, we present a tight sufficient condition for $G$ with fixed minimum degree to be $k$-connected based on the $Q$-index. Our result can be viewed as a spectral counterpart of the corresponding Dirac type condition.