A tree is called starlike if it has exactly one vertex of degree greate. than two. In [4] it was proved that two starlike trees G and H are cospectral if and only if they are isomorphic. We prove here that there exist no two non-isomorphic Laplacian cospectral starlike trees. Further, let G be a simple graph of order n with vertex set V(G) : {1,2, …, n} and let H = {$H_1$, $H_2$, …, $H_{n}$} be a family of rooted graphs. According to [2], the rooted product G(H) is the graph obtained by identifying the root of $H_{i}$ with the i-th vertex of G. In particular, if H is the family of the paths $P_k_1,P_k_2,...P_k_2$ with the rooted vertices of degree one, in this paper the corresponding graph G(H) is called the sunlike graph and is denoted by G($k_1,k_2,...k_n$). For any $(x_1,x_2,...,x_n);in;{I_*}^n$, where $I_{*}$ = : {0,1}, let G$(x_1,x_2,...,x_n)$ be the subgraph of G which is obtained by deleting the vertices $i_1,i_2,...i_j;in;V(G);(Oleq jleq n)$, provided that $x_i_1=x_i_2=...=x_i_j=o.;Let ;G[x_1,x_2,...x_n]$ be characteristic polynomial of G$(x_1,x_2,...,x_n)$, understanding that G[0,0,...,0] $equiv$1. We prove that $G[k_1,k_2,...,k_n]-sum_{xin In}[{prod_{imath=1}}^n;P_k_i+x_i-2(lambda)](-1)...G[x_1,x_2,...,X_n]$ where x=($x_1,x_2,...,x_n$);G[$k_1,k_2,...,k_n$] and $P_n(lambda)$ denote the characteristic polynomial of G($k_1,k_2,...,k_n$) and $P_n$, respectively. Besides, if G is a graph with $lambda_1(G);geq1$ we show that $lambda_1(G);leq;lambda_1(G(k_1,k_2,...,k_n))