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.



Non-regular cospectral graphs with same degree sequences
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.


