# Graph spectra and topology

This is a somewhat vague question, but I'm wondering if there has been any research into connections between the spectrum of a graph and some notion of the "topology" of that graph.

To give an example of what I mean, say we take some two-manifold $M$ and consider a graph $G$ which lives on it. By this I mean, let $G$ consist of a very large number of points on $M$ with edges connecting adjacent points. Let $A$ be the adjacency matrix of $G$. Is there some way to define a matrix $B$ dependent on $A$ such that if $M$ has trivial fundamental group the smallest eigenvalue of $B$ is negative but if $M$ has nontrivial fundamental group the smallest eigenvalue of $B$ is positive?

That was a pretty specific example - I'm really just wondering more generally if there's a way to associate a matrix with a graph such that the spectrum of the matrix "sniffs out" some topological feature of the space the graph is associated with.

• For a start a graph can embed in many different surfaces, so what matrix could we use? If the matrix depends on the surface, there's no real point in computing eigenvalues, – Chris Godsil Feb 13 '16 at 23:02
• The spectrum of the Laplacian controls the behavior of the heat equation on a graph, so loosely the eigenvalues tell you something about how quickly heat dissipates on the graph, which has something to do with how connected the graph is. More precisely, look up the Cheeger inequalities. – Qiaochu Yuan Feb 13 '16 at 23:57

A covering map of graphs $p:G\to H$ is a graph homomorphism which is locally an isomorphism, so that the neighbors of $v\in V(G)$ get mapped bijectively to the neighbors of $p(v)\in V(H)$. There are natural ways to generalize this to directed multigraphs. It's an easy exercise that every eigenvalue of $H$ will be an eigenvalue of $G$ in this situation, and moreover eigenfunctions of $G$ can be obtained from eigenfunctions of $H$ by precomposing with $p$ (or to be strict the vertex component of $p$).