Integral Laplacian graphs with a unique double Laplacian eigenvalue, I

TL;DR

This paper characterizes certain Laplacian spectra of graphs with a unique double eigenvalue, extending previous work on Laplacian realizable sets and exploring related spectral conjectures.

Contribution

It provides a complete description of graphs with specific Laplacian spectra involving a double eigenvalue and discusses related spectral conjectures.

Findings

Characterization of graphs with spectra missing two specific eigenvalues

Complete description of graphs with spectra missing the largest eigenvalues

Analysis of the $S_{n,n}$-conjecture and related spectral conjectures

Abstract

The set $S_{i,n}=\{0,1,2,\ldots,n-1,n\}\setminus\{i\}$, $1\leqslant i\leqslant n$ is called Laplacian realizable if there exists an undirected simple graph whose Laplacian spectrum is $S_{i,n}$. The existence of such graphs was established by S. Fallat et al. in 2005. In this paper, we investigate graphs whose Laplacian spectra have the form $$ S_{\{i,j\}_{n}^{m}}=\{0,1,2,\ldots,m-1,m,m,m+1,\ldots,n-1,n\}\setminus\{i,j\},\qquad 0<i<j\leqslant n, $$ and completely describe those ones with $m=n-1$ and $m=n$. We also show close relations between graphs realizing $S_{i,n}$ and $S_{\{i,j\}_{n}^{m}}$, and discuss the so-called $S_{n,n}$-conjecture and the correspondent conjecture for $S_{\{i,n\}_{n}^{m}}$.