A note on eigenvalues and Hamiltoinan properties of $k$-connected graphs

TL;DR

This paper establishes optimal upper bounds on spectral parameters of k-connected graphs to determine Hamiltonian and traceability properties, extending previous results and providing new spectral criteria for graph connectivity features.

Contribution

It introduces the best possible upper bounds of spectral radii for k-connected graphs to be Hamiltonian-connected, homogeneously traceable, and other properties, advancing spectral graph theory.

Findings

Derived optimal upper bounds for spectral radius $\lambda_1(G)$ for Hamiltonian-connectedness.

Established best possible bounds of Laplacian spectral radius $\mu_1(G)$ for various connectivity properties.

Extended previous bounds to more general classes of k-connected graphs.

Abstract

Let $\lambda_{1}(G)$ and $\mu_{1}(G)$ denote the spectral radius and the Laplacian spectral radius of a graph $G$, respectively. Li in [Electronic J. Linear Algebra 34 (2018) 389-392] proved sharp upper bounds of $\lambda_{1}(G)$ based on the connectivity to assure a connected graph to be Hamiltonian and traceable, respectively. In this paper, we present best possible upper bounds of $\lambda_{1}(G)$ for $k$-connected graphs to be Hamiltonian-connected and homogeneously traceable, respectively. Furthermore, best possible upper bounds of $\mu_{1}(G)$ to predict $k$-connected graphs to be Hamiltonian-connected, Hamiltonian and traceable are originally proved, respectively.