مشخصات پژوهش

For a graph G with vertex set fv1; : : : ; vng, let P(G) be an n × n matrix whose (i; j)-entry is the maximum number of internally disjoint vivj-paths in G, if i ̸= j, and zero otherwise. The sum of absolute values of the eigenvalues of P(G) is called the path energy of G, denoted by PE. We prove that PE of a connected graph G of order n is at least 2(n − 1) and equality holds if and only if G is a tree. Also, we determine PE of a unicyclic graph of order n and girth k, showing that for every n, PE is an increasing function of k. Therefore, among unicyclic graphs of order n, the maximum and minimum PE-values are for k = n and k = 3, respectively. These results give affirmative answers to some conjectures proposed in MATCH Commun. Math. Comput. Chem. 79 (2018) 387{398.