Sufficient Conditions for Hamiltonian Properties of Graphs Based on Quasi-Laplacian Energy

Abstract

Seeking and establishing sufficient conditions to ensure Hamiltonicity of connected graphs is crucial and valuable because of the classical NP-complete attribute. Quasi-Laplacian energy, a graph invariant in terms of the quasi-Laplacian spectrum, is a powerful tool in the resolution process of Hamilton-related problems. Let $G$ be an $n$-vertex connected graph with quasi-Laplacian eigenvalues ${\mu _1} \ge {\mu _2} \ge  \cdots  \ge {\mu _n} \ge 0$. The quasi-Laplacian energy of $G$ is defined as ${E_Q}(G) = \sum\limits_{i = 1}^n {\mu _i^2} $. In this paper, we suggest some sufficient conditions in terms of ${E_Q}(G)$ for graphs to be $k$-hamiltonian, Hamiltonian, $k$-leaf-connected, Hamilton-connected and $k$-connected, respectively.

Keywords: Quasi-Laplacian energy, Hamiltonian properties, $k$-leaf-connected, $k$-connected