On minimizing the Wiener index of unicyclic graphs with fixed girth and given degree sequence

TL;DR

This paper investigates how to construct unicyclic graphs with fixed girth and degree sequence that minimize the Wiener index, proposing three candidate structures based on the graph's centroid location.

Contribution

It introduces three specific graph structures as candidates for minimizing the Wiener index under given constraints, expanding understanding of optimal unicyclic graphs.

Findings

Identifies three candidate graphs for Wiener index minimization

Provides criteria based on centroid location for selecting the optimal structure

Enhances methods for designing graphs with minimal Wiener index

Abstract

The Wiener index of a graph is the sum of all the distances between any pair of vertices. We aim to describe graphs which minimize the Wiener index among all unicyclic graphs with fixed girth and given degree sequence. Depending on where the centroid of the graph is, we will present three candidates for the minimization, namely the greedy unicyclic graph, the cycle-centered graph and the out-greedy unicyclic graph.