Q:

eigenvalue of a graph

What does the eigenvalue of a graph mean?
I know how to compute the eigenvalues from the adjacency matrix representation of a graph but am interested in its physical significance.
If two graphs have different eigenvalues then they cannot be isomorphic. is there any other use cases of eigenvalues in the graph.

A:

If a $d$-regular graph $G$ is such that the second-largest eigenvalue $\lambda$ of $A(G)$ is significantly smaller than $d$ i.e., $d-\lambda = \Omega(1)d$, then the graph is a good expander--all sets $S$ with no more than half the number of vertices in them have $\Omega(|S|)$ neighbours outside. So to bisect the graph, many vertices would have to be removed, even for $d =3$. The dual properties of being both sparse and having high bisection width mmakes these graphs of interest to computer scientists in designing good algorithms.  

