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

Abstract

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