Path matrix and path energy of graphs

摘要

Given a graph G, we associate to it 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 Shikare et al. (2018) 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.