Extremal values for the spectral radius of the normalized distance Laplacian

TL;DR

This paper investigates the extremal spectral radii of the normalized distance Laplacian in graphs, proving the complete graph has the minimum spectral radius and providing partial results for the maximum case.

Contribution

It proves Reinhart's conjecture that the complete graph uniquely minimizes the spectral radius and advances understanding of the maximum spectral radius in this context.

Findings

Complete graph is the unique minimizer of spectral radius.

Partial results towards identifying the maximizer of spectral radius.

Established bounds and structural properties related to spectral extremal values.

Abstract

The normalized distance Laplacian of a graph $G$ is defined as $\mathcal{D}^\mathcal{L}(G)=T(G)^{-1/2}(T(G)-\mathcal{D}(G))T(G)^{-1/2}$ where $\mathcal{D}(G)$ is the matrix with pairwise distances between vertices and $T(G)$ is the diagonal transmission matrix. In this project, we study the minimum and maximum spectral radii associated with this matrix, and the structures of the graphs that achieve these values. In particular, we prove a conjecture of Reinhart that the complete graph is the unique graph with minimum spectral radius, and we give several partial results towards a second conjecture of Reinhart regarding which graph has the maximum spectral radius.