Abstract

Let G be a graph with the vertex set fv1; : : : ; vng. The path matrix P (G) is
an n × n matrix whose (i; j)entry is the maximum number of internally disjoint
vivjpaths in G, if i 6= j, and zero otherwise. The sum of absolute values of the
eigenvalues of P (G) is called the path energy of G. In this paper the path energy
of bicyclic graphs are investigated. In particular, among bicyclic graphs of a fxed
order, the graphs with maximum and minimum path energy are characterized.
Using these results, we provide afrmative answers to some conjectures proposed
in MATCH Commun. Math. Comput. Chem. 79 (2018) 387–398.
