MMN-808
Extremal unicyclic graphs with respect to additively weighted Harary index
Abstract
In this paper we define cycle-star graph $CS_{k,n−k}$ to be a graph on $n$ vertices consisting
of the cycle of length $k$ and $n−k$ leafs appended to the same vertex of the cycle. Also, we
define cycle-path graph $CP_{k,n−k}$ to be a graph on $n$ vertices consisting of the cycle of length
$k$ and of path on $n−k$ vertices whose one end is linked to a vertex on a cycle. We establish
that cycle-star graph $CS_{3,n−3}$ is the only maximal graph with respect to additively weighted
Harary index among all unicyclic graphs on $n$ vertices, while cycle-path graph $CP_{3,n−3}$ is the
only minimal unicyclic graph (here $n$ must be at least 5). The values of additively weighted
Harary index for extremal unicyclic graphs are established, so these values are the upper and the lower bound for the value of additively weighted Harary index on the class of unicyclic
graphs on $n$ vertices.
Vol. 16 (2015), No. 2, pp. 1163-1180