Integral Laplacian graphs with a unique double Laplacian eigenvalue, II

TL;DR

This paper characterizes and describes the structure of simple connected graphs whose Laplacian spectra are missing exactly two specific eigenvalues, focusing on cases where the spectrum set has particular forms with m=1,2.

Contribution

It provides a complete description of graphs realizing Laplacian spectra with two missing eigenvalues for specific spectrum sets, expanding understanding of Laplacian spectral realizability.

Findings

Graphs realizing the sets $S_{\{i,j\}_{n}^{m}}$ with $m=1,2$ are fully characterized.

The structure of these graphs is explicitly determined.

The results extend the classification of Laplacian spectra with unique eigenvalue properties.

Abstract

The set $S_{\{i,j\}_{n}^{m}}=\{0,1,2,\ldots,m-1,m,m,m+1,\ldots,n-1,n\}\setminus\{i,j\},\quad 0<i<j\leqslant n$, is called Laplacian realizable if there exists a simple connected graph $G$ whose Laplacian spectrum is $S_{\{i,j\}_{n}^{m}}$. In this case, the graph $G$ is said to realize $S_{\{i,j\}_{n}^{m}}$. In this paper, we completely describe graphs realizing the sets $S_{\{i,j\}_{n}^{m}}$ with $m=1,2$ and determine the structure of these graphs.