The largest $n-1$ Hosoya indices of unicyclic graphs

TL;DR

This paper identifies and orders the unicyclic graphs with the largest Hosoya indices, extending previous research and providing a clear hierarchy of these graphs based on their edge-independent set counts.

Contribution

The paper determines the largest $n-1$ unicyclic graphs by Hosoya index, advancing the understanding of extremal properties in unicyclic graphs.

Findings

Identified the unicyclic graphs with the maximal Hosoya index.

Established an ordering of unicyclic graphs based on Hosoya indices.

Extended previous research on extremal unicyclic graphs.

Abstract

The Hosoya index $Z (G)$ of a graph $G$ is defined as the total number of edge independent sets of $G$. In this paper, we extend the research of [J. Ou, On extremal unicyclic molecular graphs with maximal Hosoya index, \textit{Discrete Appl. Math.} 157 (2009) 391--397.] and [Y. Ye, X. Pan, H. Liu, Ordering unicyclic graphs with respect to Hosoya indices and Merrifield-Simmons indices, \textit{MATCH Commun. Math. Comput. Chem.} 59 (2008) 191--202.] and order the largest $n-1$ unicyclic graphs with respect to the Hosoya index.