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The eigenvalues of $G$ determine the eigenvalues of $\tilde G$ in the regular case. If I calculated correctly, just replace the eigenvalue $d$ by the two zeros of $x^2-dx-n$, and keep the other eigen …

$\lambda_n = k - \mu_1$, where $\mu_1$ is the least (most negative) eigenvalue of the adjacency matrix. That eigenvalue is well studied, so that is where to search. For example $\mu_1\le -1$, hence …

As far as I know, the computation of these values up to 11 vertices by van Dam and Haemers is still the best result. No asymptotics are known.

If $k$ is the greatest common divisor of the cycle lengths of the digraph, then the spectrum is invariant under rotation around the origin by $2\pi/k$. This is an application of the Perron-Frobenius t …

Chris seems to have forgotten that he and I published a generalization of the NEPS in C. D. Godsil and B. D. McKay, Constructing cospectral graphs, Aequationes Mathematicae, 25 (1983) 257-268. This co …

These questions are all answered in the Wikipedia article Johnson graph. As Chris noted, it doesn't matter if you consider the adjacency matrix or the Laplacian matrix. The eigenvectors stay the same …

This is expander territory and someone will doubtless give a reference soon. Meanwhile, here's a simple proof that $\liminf h_n \le 1$. Consider a connected induced subgraph $H$ with $n_1,n_2,n_3$ v …

Yes, and you can find many more than $\log n$ of them. Take an expander $G$ (for example a random cubic graph). Take $k$ copies, where $k=o(n^{1/2})$. Now randomly relabel each copy. The probability o …

The moments (power symmetric functions, sums of powers of the roots of) the characteristic polynomial enumerate all closed walks in the graph. Chris Godsil proved that the moments of the matching pol …

If you take an infinite regular tree of degree $d$ and fix one vertex $v$, then the number of closed walks of length $2k$ (there are none of odd length) starting at $v$ grows like $4^k(d-1)^k$ as $k\t …

Take the case of choosing edges independently with probability $p=n^{-2+\epsilon}$. As you say, it won't make much difference compared to choosing $n^{1+\epsilon}$ edges. Assume $\epsilon<\frac12$. …

You are asking for relationships between the maximum independent set and the eigenvalues. If you search with those terms you will find several. For example, Haemers proved that the maximum size of a …

The smallest eigenvalue can go up or down when an edge is removed. For "down": $G=K_n$ for $n\ge 3$. For "up": Take $K_n$ for $n\ge 1$ and append a new vertex attached to a single vertex of the orig …

The moments of the adjacency matrix eigenvalues count closed walks in the graph, while the moments of the matching polynomial roots count tree-like closed walks. When the graph has few short cycles, a …

This paper says it is NP-hard and gives three references.