Alternative Proofs on the Indices of Cacti and Unicyclic Graphs with $n$ Vertices

TL;DR

This paper provides alternative proofs for the maximal spectral radius of specific cactus and unicyclic graphs with n vertices, highlighting their uniqueness among their respective graph classes.

Contribution

It offers new proofs confirming the extremal spectral radius properties of certain cactus and unicyclic graphs, enhancing understanding of their spectral characteristics.

Findings

$H_n$ has the maximal spectral radius among all cacti with $n$ vertices.

$K_{1,n-1}^+$ has the maximal spectral radius among all unicyclic graphs with $n$ vertices.

$H_n$ is the unique odd-cycle graph with the maximal spectral radius among all odd-cycle graphs with $n$ vertices.

Abstract

Let $H_n$ be the cactus obtained from the star $K_{1,n-1}$ by adding $\lfloor \frac{n-1}{2}\rfloor$ independent edges between pairs of pendant vertices. Let $K_{1,n-1}^+$ be the unicyclic graph obtained from the star $K_{1,n-1}$ by appending one edge. In this paper we give alternative proofs of the following results: Among all cacti with $n$ vertices, $H_n$ is the unique cactus whose spectral radius is maximal, and among all unicyclic graphs with $n$ vertices, $K_{1,n-1}^+$ is the unique unicyclic graph whose spectral radius is maximal. We also prove that among all odd-cycle graphs with $n$ vertices, $H_n$ is the unique odd-cycle graph whose spectral radius is maximal.