Extremal unicyclic graphs with respect to additively weighted Harary index

TL;DR

This paper characterizes extremal unicyclic graphs with respect to the additively weighted Harary index, identifying the unique maximal and minimal structures and providing their index values.

Contribution

It introduces specific extremal unicyclic graphs and determines their additively weighted Harary index, establishing bounds for all such graphs.

Findings

Cycle-star graph CS3,n-3 is the unique maximal unicyclic graph.

Cycle-path graph CP3,n-3 is the unique minimal unicyclic graph.

Values of the index for extremal graphs serve as bounds for all unicyclic graphs.

Abstract

In this paper we define cycle-star graph CSk,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 CPk,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 CS3,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 on n vertices.