Faber-Krahn type inequality for unicyclic graphs

TL;DR

This paper extends the Faber-Krahn inequality to unicyclic graphs, demonstrating that among graphs with a fixed degree sequence, certain configurations minimize the first Dirichlet eigenvalue, similar to the continuous case.

Contribution

It establishes a Faber-Krahn type inequality for unicyclic graphs with fixed degree sequences, a novel extension from the classical continuous domain to discrete graph structures.

Findings

Faber-Krahn inequality holds for unicyclic graphs with given degree sequences.

Identifies conditions under which the inequality is valid for these graphs.

Provides insights into spectral properties of unicyclic graphs.

Abstract

The Faber-Krahn inequality states that the ball has minimal first Dirichlet eigenvalue among all bounded domains with the fixed volume in $\mathbb{R}^n$. In this paper, we investigate the similar inequality for unicyclic graphs. The results show that the Faber-Krahn type inequality also holds for unicyclic graphs with a given graphic unicyclic degree sequence with minor conditions.