Toughness, hamiltonicity and spectral radius in graphs

TL;DR

This paper explores spectral conditions related to toughness that guarantee the existence of Hamiltonian cycles in graphs, extending classical results and providing new spectral criteria for toughness and Hamiltonicity.

Contribution

It introduces spectral conditions involving spectral radius that ensure a graph's toughness and Hamiltonicity, extending existing theorems and addressing Chvátal's conjecture from a spectral perspective.

Findings

Spectral radius condition for Hamiltonicity in 1-tough graphs

Extension of toughness bounds using spectral radius and eigenvalues

Characterization of graphs with Hamiltonian cycles based on spectral properties

Abstract

The study of the existence of hamiltonian cycles in a graph is a classic problem in graph theory. By incorporating toughness and spectral conditions, we can consider Chv\'{a}tal's conjecture from another perspective: what is the spectral condition to guarantee the existence of a hamiltonian cycle among $t$-tough graphs? We first give the answer to $1$-tough graphs, i.e. if $\rho(G)\geq\rho(M_{n})$, then $G$ contains a hamiltonian cycle, unless $G\cong M_{n}$, where $M_{n}=K_{1}\nabla K_{n-4}^{+3}$ and $K_{n-4}^{+3}$ is the graph obtained from $3K_{1}\cup K_{n-4}$ by adding three independent edges between $3K_{1}$ and $K_{n-4}$. The Brouwer's toughness theorem states that every $d$-regular connected graph always has $t(G)>\frac{d}{\lambda}-1$ where $\lambda$ is the second largest absolute eigenvalue of the adjacency matrix. In this paper, we extend the result in terms of its spectral radius, i.e. we provide a spectral condition for a graph to be 1-tough with minimum degree $\delta$ and to be $t$-tough, respectively.