Integral Laplacian graphs with a unique repeated Laplacian eigenvalue, I

Abstract

Abstract The set S i , n = { 0 , 1 , 2 , … , n − 1 , n } \ { i } {S}_{i,n}=\left\{0,1,2,\ldots ,n-1,n\right\}\setminus \left\{i\right\} , 1 ⩽ i ⩽ n 1\leqslant i\leqslant n , is called Laplacian realizable if there exists an undirected simple graph whose Laplacian spectrum is S i , n {S}_{i,n} . The existence of such graphs was established by Fallat et al. (On graphs whose Laplacian matrices have distinct integer eigenvalues, J. Graph Theory 50 (2005), 162–174). In this article, we consider graphs whose Laplacian spectra have the form S { i , j } n m = { 0 , 1 , 2 , … , m − 1 , m , m , m + 1 , … , n − 1 , n } \ { i , j } , 0 < i < j ⩽ n , {S}_{{\left\{i,j\right\}}_{n}^{m}}=\left\{0,1,2,\ldots ,m-1,m,m,m+1,\ldots ,n-1,n\right\}\setminus \left\{i,j\right\},\hspace{1.0em}0\lt i\lt j\leqslant n, and completely describe those with m = n − 1 m=n-1 and m = n m=n . We also show close relations between graphs realizing S i , n {S}_{i,n} and S { i , j } n m {S}_{{\left\{i,j\right\}}_{n}^{m}} and discuss the so-called S n , n {S}_{n,n} -conjecture and the corresponding conjecture for S { i , n } n m {S}_{{\left\{i,n\right\}}_{n}^{m}} .