Quasi-Laplacian energy of \(\psi \)-sum graphs

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 \(\Gamma \) $\mathrm{\Gamma }$ be an n-vertex connected graph with quasi-Laplacian eigenvalues \(\mu _{1}\geqslant \mu _{2}\geqslant \cdots \geqslant \mu _{n}\geqslant 0\) ${\mu }_1⩾{\mu }_2⩾\cdots ⩾{\mu }_n⩾0$ . The quasi-Laplacian energy of \(\Gamma \) $\mathrm{\Gamma }$ is defined as \(E_Q\left( \Gamma \right) =\sum _{i=1}^n{\mu _{i}^{2}}\) $E_Q\left(\mathrm{\Gamma }\right)={\sum }_{i=1}^n{\mu }_i^2$ . The \(\psi \) $\psi$ -sum graphs are generated by utilizing Cartesian product operation for \(\psi (\Gamma _1)\) $\psi ({\mathrm{\Gamma }}_1)$ and \(\Gamma _2\) ${\mathrm{\Gamma }}_2$ , denoted by \(\Gamma _1+_{\psi }\Gamma _2\) ${\mathrm{\Gamma }}_1{+}_{\psi }{\mathrm{\Gamma }}_2$ . In this paper, in terms of quasi-Laplacian energy of factor graphs, we characterize the quasi-Laplacian energy of four kinds of \(\psi \) $\psi$ -sum graphs. As applications, we determine the quasi-Laplacian energy of several special \(\psi \) $\psi$ -sum graphs generated by base graphs, i.e., path, cycle, complete graph and complete bipartite graph, respectively.