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

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 $(\alpha ,\beta )\mathrm{\ne }(0,0)$ 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 $\alpha \ge \beta >0$, we present sufficient conditions for the Hamiltonian and traceable graphs.