MMN-808

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

DOI: 10.18514/MMN.2015.808

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