There are some facts that can be found by the spectrum of adjacency matrix of graph.For example, the number of edges and vertices, is bipartite or not, is complete multipartite or not and so on. Can we say anything about the clique number of two cospectral graphs?

We can construct the graphs $G_1$ and $G_2$ that they are cospectral and for arbitrary $k\in N$, the difference of clique number of these two graphs be grater than $k$. But, as I know, these graphs are disconnected.

So, my question is in connected case. I mean, suppose $G_1$ and $G_2$ are cospectral and connected. What can we say about their clique numbers?

Thanks for any helpful answer.