Eigenvalues of Adjacency Matrix are Integer?

Question

There are some related discussions here:
Computation of Eigenvalues of Adjacency Matrix of Cycle
Eigenvalues of adjacent matrix
eigenvalue of adjacency matrix of bipartite graph

However, is there a way to prove that eigenvalues of adjacency matrix of any graph are integer?

The graph should obey

1. connected
2. undirected
3. edges weight are $1$

I know the trace of adjacency matrix is $0$. However, is there intuitive way or formal proof to say this?

Answer 1

Here is a counterexample for your conjecture. Let $p, q$ be distinct primes, and consider the graph $K_{p,q}$. We note that the eigenvalues of $K_{p, q}$ are $\pm \sqrt{pq}$ and $0^{pq-2}$. Now $\sqrt{pq}$ is certainly not an integer.