# Non-regular cospectral graphs with same degree sequences

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.