Spectral radius of a star with one long arm

TL;DR

This paper investigates the spectral properties of a specific class of starlike trees, deriving bounds for their largest eigenvalues and establishing relationships between different configurations.

Contribution

It provides new bounds for the spectral radius of starlike trees with one long arm and multiple short arms, and relates their eigenvalues to those of other starlike trees.

Findings

Largest eigenvalue bounds: rom rom rom

Eigenvalue equality between different starlike trees

Spectral radius insights for starlike trees

Abstract

A tree is said to be starlike if exactly one vertex has degree greater than two. In this paper, we will study the spectral properties of $S(n,k \cdot 1)$, that is, the starlike tree with $k$ branches of length 1 and one branch of length $n$. The largest eigenvalue $\lambda_1$ of $S(n,k \cdot 1)$ satisfies $\sqrt{k+1} \leq \lambda_1 < k/\sqrt{k-1}$. Moreover, the largest eigenvalue of $S(n,k \cdot 1)$ is equal to the largest eigenvalue of $S(k \cdot (n+1) )$, which is the starlike tree that has $k$ branches of length $n-1$. Using the spectral radii of $S(n,k \cdot 1)$ we can show