Extremal unicyclic graphs with respect to additively weighted Harary index

Abstract

In this paper we define cycle-star graphCSk;n k to be a graph onn vertices consisting of the cycle of lengthk andn k leafs appended to the same vertex of the cycle. Also, we define cycle-path graphCPk;n k to be a graph onn vertices consisting of the cycle of lengthk and of path on n k vertices whose one end is linked to a vertex on a cycle. We establish that cycle- star graphCS3;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 CP3;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 onn vertices. 2010 Mathematics Subject Classification: 05C35