Path matrix and path energy of graphs

TL;DR

This paper introduces a path matrix for graphs, resolves four conjectures on path energy, and presents an efficient algorithm for computing the path matrix, advancing understanding of graph path properties.

Contribution

It resolves four conjectures on path energy and provides an efficient algorithm for computing the path matrix in graphs.

Findings

Resolved four conjectures on path energy.

Developed an $O(|E| |V|^3)$ algorithm for path matrix computation.

Enhanced understanding of graph path properties.

Abstract

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, \emph{On Path Eigenvalues and Path Energy of Graphs}, MATCH Commun. Math. Comput. Chem. {\bf 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.