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I think your conjecture does not fly. Indeed, consider a "Paley tournament": let $p=4k-1$ e a prime, and define a digraph with the set of vertices 0,1,..,$p-1$, so that $i$ is connected to $j$ wheneve …

To rectify joro's answer; he is certainly correct about the general direction, although it is not clear from his answer whether any of these 35-vertex SRGs is Cayley. However, on 25 vertices there ar …

This (i.e. the non-increase of $\lambda_1$ w.r.t. edge removal) follows from the fact that $$\lambda_1=\max_{x\geq 0,\|x\|=1} x^\top A x.$$ The latter is a consequence of the characterisation of $\la …

There exist co-spectral trees. As there are no cycles to preserve, they fit the bill. As far as 2-connected graphs are concerned, I would try finding a 2-connected graph $\Gamma$ which can be obtaine …

The automorphism group of this graph is $S_{c-1}$. Note that the vertex 1 in your clique cannot be moved anywhere (look at the degrees). On the other hand, a permutation of the remaining vertices in { …

The dimension of the nullspace $N$ of $D-A$ is the number of the connected components of the graph, and moreover one can find an orthogonal basis of $N$ with non-negative eigenvectors, each of them be …

The graph $\Omega(n)$ is one of the relations of the Hamming association scheme $H(n,2)$, namely, the one corresponding to the Hamming distance $n/2$, see e.g. here. Its eigenvalues are given by the v …

Such graphs are studied a lot in coding theory and in the theory of association schemes. In paricular the eigenvalues can be explicitly written down in terms of Krawchuk polynomials $K_k(u)$. You c …

I believe that for $p$ prime $DB(m,p)$ has one 0 eigenvalue (this is easy to check), and the other eigenvalues are all equal to $p$. This is something that we observed while working on https://arxiv.o …

Permutation graphs are perfect, therefore Lovasz theta, which is essentially a spectral bound, computes the clique number in polynomial time in polynomial time for this class of graphs.

Computing Lovasz $\theta$ for circulant graphs can be reduced to linear programming; this is well-known, I think (already mentioned in A.Schrijver's 1979 paper "A comparison of the Delsarte and Lovasz …

Distance matrix is not going to help, as examples of non-isomorphic co-spectral strongly regular graphs would tell you. The smallest examples like this exist on 16 vertices: Shrikhande graph. In gene …

Yes, there are lots of such examples of strongly regular graphs. E.g. Chang graphs give an example of for 1). For 2), many cospectral graphs have trivial automorphism groups (thus isomorphic as groups …

Here is the link to paper in question: On the normalized Laplacian eigenvalues of graphs. Ars Comb. Unfortunately only abstract seems to be available to the public Let G = (V, E) be a simple connected …