I am looking for a large family (infinite pairs) of cospectral graphs with these condtions:
The graphs are non-regular,
Minimum degree is greater than $1$,
The degree sequences of these cospectral graphs are the same.
I need cospectrality by adjacency matrix and the graphs are simple.
The motivation for asking this question is that; if we have counterexample for reconstruction conjecture, the two graphs have these properties. I want to study such family of graphs.
Let $D$ be a Steiner triple system on $v$ points. (So $v\equiv1,3$ mod 6). The incidence graph is the bipartite graph with the $v$ points as one colour class and the $v(v-1)/6$ blocks as the second; a point is incident with the $(v-1)/2$ blocks that contain it.
Let $N$ be the point-block incidence matrix of the system. Then $NN^T=\frac12(v-3)I+J$ and $NN^T$ has the same non-zero eigenvalues as $NN^T$, with the same multiplicities. The adjacency matrix $A$ has the form $$A =\begin{pmatrix}0&N\\ N&0\end{pmatrix}$$ and $$A^2 =\begin{pmatrix}NN^T&0\\ 0&N^TN\end{pmatrix}$$ from which it follows that spectrum of $A$ is determined by $v$.
The incidence graphs have degree set $\{3,(v-1)/2\}$, so these graphs are not regular if $v\ge9$. There are 80 Steiner triple systems on 15 points and Kaski and Ostergard showed that there are 11,084,874,829 on 19 points.
The number of walks of length $k$ in an incidence graph is determined by $v$ (exercise) and it follows that the complements of the incidence graphs are also cospectral.
