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.
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Acknowledgements
The authors would like to express their sincere gratitude to all reviewers for valuable suggestions, which are helpful in improving and clarifying the original manuscript. We thank the National Institute of Education, Nanyang Technological University, where part of this research was performed. This work was partly supported by the National Natural Science Foundation of China (Nos. 61977016, 61572010), Natural Science Foundation of Fujian Province (Nos. 2023J01539, 2020J01164). This work was also partly supported by Fujian Alliance of Mathematics (No. 2023SXLMMS04) and China Scholarship Council (No. 202108350054).
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Zhuo, Y., Zhou, S. & Yang, L. Quasi-Laplacian energy of \(\psi \)-sum graphs. J. Appl. Math. Comput. 70, 535–550 (2024). https://doi.org/10.1007/s12190-023-01976-3
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DOI: https://doi.org/10.1007/s12190-023-01976-3