Let
![$\mathcal{G}(\Delta^{x_{\Delta}},\ldots, 2^{x_{2}},1^{x_{1}})$](img25.png)
be the set of connected graph with degree sequence
![$\ldots,2^{x_{2}},1^{x_{1}})$](img27.png)
. In this paper,
we show that the degree sequences of
![$\mathcal{G}(3^{n_3},2^{n_2},1^{n_1})$](img28.png)
and
![$\mathcal{G}(4^{n_4},2^{n_2},1^{n_1})$](img29.png)
are determined by Laplacian
spectrum with some restrictions. By those results we obtain that
sun graph [4],
![$C_{n}\circ2K_{1}$](img30.png)
[6]
and
![$\mathcal{G}(4^1,2^{n-3},1^2)$](img31.png)
are determined by their Laplacian
spectra. Furthermore, we prove that any graph
![$G\in\mathcal{G}(\Delta^{1},2^{n_2},1^{n_1})~(\Delta\ge 3)$](img32.png)
, whose
degree sequence is determined by Laplacian spectrum except that
![$G$](img33.png)
is a bicyclic graph with
![$\Delta=4$](img34.png)
. Moreover, if
![$G$](img33.png)
is a bicyclic graph with
![$\Delta=4$](img34.png)
, then
![$G$](img33.png)
may be
![$L$](img35.png)
-cospectral to a graph with degree
sequence
![$(3^{3},2^{n_{2}-3}, 1^{n_{1}+1})$](img36.png)
. Applying this result
we conclude that some graphs such as
starlike
trees[22],
unicyclic graph ![$U_{n,p}$](img37.png)
[27],
friendship graph and
butterfly graph et al. are
determined by their Laplacian spectra. Moreover, we give a
Laplacian spectral characterization of the degree sequence of
![$p$](img38.png)
-rose graph which supports Liu's conjecture [19].with
![$n$](img39.png)