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.



The cliques of cospectral graphs
Let $X$ and $Y$ be two cospectral graphs with maximum clique size $a$ and $b$ respectively. Then their $k$-fold strong powers $X(k)$ and $Y(k)$ are cospectral and the
maximum size of a clique is $a^k$ and $b^k$ respectively. (The cliques of maximum size in a strong product are strong products of maximum sized cliques in the factors.)
Since the strong product of connected graphs is connected, we have our examples.


We could take $X$ and $Y$ to be the Shrikande graph ($a=3$) and the line graph of $K\_{4,4}$ ($b=4$). There are smaller examples, but I am too lazy to look right now.


