Ordering Unicyclic Graphs with Respect to F-index

TL;DR

This paper studies the F-index, which sums the cubes of vertex degrees, in unicyclic graphs, identifying those with the highest and lowest F-indices through specific transformations.

Contribution

It introduces transformations to analyze F-indices and characterizes extremal unicyclic graphs with the largest and smallest F-indices.

Findings

Identified unicyclic graphs with top five largest F-indices.

Identified unicyclic graphs with the two smallest F-indices.

Developed transformation techniques for F-index analysis.

Abstract

F-index of a graph is the sum of the cube of the degrees of the vertices. In this paper, we investigate the F-indices of unicyclic graphs by introducing some transformation, and characterize the unicyclic graphs with the first five largest F-indices and the unicyclic graphs with the first two smallest F-indices, respectively.