  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 $S_{\{i,n\}_{n}^{m}}$.
