Quasi-Laplacian energy of $\psi$-sum graphs

Abstract

As a hot issue in the field of algebraic graph theory, the quasi-Laplacian energy of a graph is a graph invariant in terms of the quasi-Laplacian spectrum, and is versatile in multidisciplinarity, such as social network analysis, theoretical computer science, mathematical chemistry, and so on. Let $\Gamma$ be an n-vertex connected graph with quasi-Laplacian eigenvalues $\mu _{1}\geqslant \mu _{2}\geqslant \cdots \geqslant \mu _{n}\geqslant 0$. The quasi-Laplacian energy of $\Gamma$ is defined as $E_Q\left( \Gamma \right) =\sum _{i=1}^n{\mu _{i}^{2}}$. The $\psi$-sum graphs are generated by utilizing Cartesian product operation for $\psi (\Gamma _1)$ and $\Gamma _2$, denoted by $\Gamma _1+_{\psi }\Gamma _2$. In this paper, in terms of quasi-Laplacian energy of factor graphs, we characterize the quasi-Laplacian energy of four kinds of $\psi$-sum graphs. As applications, we determine the quasi-Laplacian energy of several special $\psi$-sum graphs generated by base graphs, i.e., path, cycle, complete graph and complete bipartite graph, respectively.