Integral Laplacian graphs with a unique repeated Laplacian eigenvalue, I

The set S-i,S-n={0,1,2,& mldr;,n-1,n}\{i} , 1 <= i <= n , is called Laplacian realizable if there exists an undirected simple graph whose Laplacian spectrum is S-i,S-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 formS({i,j})m(n)={0,1,2,& mldr;,m-1,m,m,m+1,& mldr;,n-1,n}\{i,j}, 0 < i < j <= n,and completely describe those with m=n-1 and m=n . We also show close relations between graphs realizing S-i,S-n and S({i,j})m(n) and discuss the so-called S-n,S-n -conjecture and the corresponding conjecture for S({i,j})m(n) .