Cospectral graphs for the normalized Laplacian

Let $G(a_1,a_2,\dots ,a_k)$ be a simple graph with vertex set $V(G)=V_1\cup V_2\cup \cdots \cup V_k$ and edge set $E(G)=\{(u,v)|u\in V_i,v\in V_{i+1},i=1,2,\dots ,k-1\}$, where $|V_i|=a_i>0$ for $1\le i\le k$ and $V_i\cap V_j=\mathrm{\varnothing }$ for $i\ne j$. Given two positive integers $k$ and $n$, and $k-2$ positive rational numbers $t_2,t_3,\dots ,t_{\lceil k/2\rceil }$ and $t_2^{\prime },t_3^{\prime },\dots ,t_{\lfloor k/2\rfloor }^{\prime }$, let $\mathrm{{\rm Y}}(n;k{)}_t^{t^{\prime }}=\{G(a_1,a_2,\dots ,a_k)|\sum _{i=1}^k{a}_i=n,a_{2i-1}=t_i{a}_1,a_{2j}=t_j^{\prime }{a}_2,i=2,3,\dots ,\lceil k/2\rceil ,$ $j=2,3,\dots ,\lfloor k/2\rfloor ;t=(t_2,t_3,\dots ,t_{\lceil k/2\rceil }),t^{\prime }=(t_2^{\prime },t_3^{\prime },\dots ,t_{\lfloor k/2\rfloor }^{\prime });a_s\in N,1\le s\le k\}$, where $N$ is the set of positive integers. In this paper, we prove that all graphs in $\mathrm{{\rm Y}}(n;k{)}_t^{t^{\prime }}$ are cospectral with respect to the normalized Laplacian if it is not an empty set.