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

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.