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Abstract
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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.
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