Computing the Energy of Certain Graphs based on Vertex Status

Background: The concept of Hückel molecular orbital theory is used to compute the graph energy numerically and graphically on the base of the status of a vertex. Objective: Our aim is to explore the graph energy of various graph families on the base of the status adjacency matrix and its Laplacian version. Methods: We opt for the technique of finding eigenvalues of adjacency and Laplacian matrices constructed on the base of the status of vertices. Results: We explore the exact status sum and Laplacian status sum energies of a complete graph, complete bipartite graph, star graphs, bistar graphs, barbell graphs and graphs of two thorny rings. We also compared the obtained results of energy numerically and graphically. Conclusion: In this article, we extended the study of graph spectrum and energy by introducing the new concept of the status sum adjacency matrix and the Laplacian status sum adjacency matrix of a graph. We investigated and visualized these newly defined spectrums and energies of well-known graphs, such as complete graphs, complete bi-graphs, star graphs, friendship graphs, bistar graphs, barbell graphs, and thorny graphs with 3 and 4 cycles.