Path matrix and path energy of graphs
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Given a graph G, we associate a path matrix P whose (i, j) entry represents the maximum number of vertex disjoint paths between the vertices i and j, with zeros on the main diagonal. In this note, we resolve four conjectures from [M. M. Shikare, P. P. Malavadkar, S. C. Patekar, I. Gutman, On Path Eigenvalues and Path Energy of Graphs, MATCH Commun. Math. Comput. Chem. 79 (2018), 387--398.] on the path energy of graphs and finally present efficient O(|E| |V|^3) algorithm for computing the path matrix used for verifying computational results.
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