Integral Laplacian graphs with a unique double Laplacian eigenvalue, II

The set $S_{\{i,j{\}}_n^m}=\{0,1,2,\dots ,m-1,m,m,m+1,\dots ,n-1,n\}\setminus \{i,j\},0<i<j⩽n$, 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.