Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth

TL;DR

This paper identifies the unicyclic graphs with fixed girth that minimize the Laplacian spectral radius, providing exact conditions for when this minimum is achieved.

Contribution

It proves the unique minimizer of the Laplacian spectral radius among unicyclic graphs with fixed girth under specific size conditions.

Findings

The graph $U_{n,g}$ minimizes the Laplacian spectral radius for even girth when $n \\geq 2g-1$.

The same graph minimizes the spectral radius for odd girth when $n \\geq 3g-1$.

The paper establishes exact bounds for the minimization problem.

Abstract

In this paper we consider the following problem: Over the class of all simple connected unicyclic graphs on $n$ vertices with girth $g$ ($n,g$ being fixed), which graph minimizes the Laplacian spectral radius? We prove that the graph $U_{n,g}$ (defined in Section 1) uniquely minimizes the Laplacian spectral radius for $n\geq 2g-1$ when $g$ is even and for $n\geq 3g-1$ when $g$ is odd.