The Largest and Smallest Eigenvalues of Matrices and Some Hamiltonian Properties of Graphs

TL;DR

This paper investigates eigenvalues of a matrix family derived from a graph's degree and adjacency matrices to establish new conditions for Hamiltonian and traceable graphs.

Contribution

It introduces eigenvalue-based criteria for Hamiltonian and traceable properties using matrices defined by graph degree and adjacency.

Findings

Eigenvalues provide sufficient conditions for Hamiltonian graphs.

Eigenvalues give sufficient conditions for traceable graphs.

New spectral criteria improve understanding of graph Hamiltonicity.

Abstract

Let $G = (V, E)$ be a graph. We define matrices $M(G; \alpha, \beta)$as $\alpha D + \beta A$, where $\alpha$, $\beta$ are real numbers such that $(\alpha, \beta) \neq (0, 0)$ and $D$ and $A$ are the diagonal matrix and adjacency matrix of $G$, respectively. Using the largest and smallest eigenvalues of $M(G; \alpha, \beta)$ with $\alpha \geq \beta > 0$, we present sufficient conditions for the Hamiltonian and traceable graphs.