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This paper analyzes the weighted adjacency spectrum of complete multipartite graphs, focusing on characterization, integrality, and the effects of edge deletion. It corrects previous findings regarding energy and spectral radius changes upon edge deletion in complete graphs with weighted matrices. The study also provides counterexamples and corrections related to the $ISI$ energy of regular tripartite graphs and settles an open problem concerning $ISI$ energy changes in multipartite graphs.
Edge deletion doesn't always increase energy in weighted graphs, overturning prior claims and demanding a re-evaluation of spectral graph theory assumptions.
The article presents weighted adjacency spectrum of complete multipartite graphs, characterize its families with three distinct eigenvalues and identifies integral matrices. Also, we observe that for almost all weighted matrices, the energy and the spectral radius of a complete graph decreases upon edge deletion, thereby correcting and refining earlier published results in [Bilal and Munir, Int. J. Quantum Chem. (2024)]. Furthermore, we give counter examples related to $ISI$ energy decrease of regular tripartite graph by edge deletion and give its correct $ISI$ spectrum and $ISI$ energy and settle an open problem related to $ISI$ energy change of the multipartite graph. Also, we calculate the weighted adjacency spectrum of crown multipartite graph and discuss its integral spectral weighted spectrum.
