Can somebody help me to construct two family of finite simple connected graph $G_i$ and $H_i$, $i=1, 2, \cdots,n$ ($n$ possibly large), such that:
$1)$ $G_i\ncong H_i$ for $i=1, 2, \cdots, n$
$2)$ $|V(G_i)|=|V(H_i)|, |E(G_i)|=|E(H_i)|$
$3)$ If $C_k(G)$ denotes the number of closed walk of length $k$ in graph $G$, we have:
$C_k(G_i)=C_k(H_i)$ for $i=1, 2, \cdots, n$
$4)$ Preferably, I need these graphs be $a)$minimal and $b)$highly irregular(or has one of these two conditions $(a)$ or $(b)$).
$Definition 1:$ A graph $G$ is Highly irregular, if every vertex $v$ of $G$ is adjacent only to vertices with distinct degrees.
$Definition 2:$ The sequence of graphs $G_i$,$i=1,2,\cdots,n$, is minimal, if the number of vertices of every $G_i$ is minimum.
For example, two trees $T_1$ and $T_2$ with degree sequences $4,4,1,1,1,1,1,1$ and $5,2,2,1,1,1,1,1$ respectively, are minimal, because they are minimum vertices co-spectral trees.
I will appreciate any help and guidance.