I know that this question is some sort of bridge between Informatics and Mathematics, not knowing the best place where to post this question, I opted for this place because of the type of answer I want (which concerns Math more than anything).
Consider to have a graph represented by a collection of nodes and connections. The best way to describe the graph is through the adjacency matrix (AM): a matrix that has 0 or 1 if one node is connected to another (we consider non-directed graphs, so we have all bidirectional connections).
Does anyone know whether the eigenvalues of the matrix implies something for the graph??? properties, topology.... I ask this question because I've almost finished studying Markov Chains. In a chain, the matrix of transition probabilities P (for discrete time markov chains), or the matrix of transition frequencies Q (for cont-time markov chains), can be evaluated (their eigenvalues) in order to inspect whether the chain is ergodic or not (with a parallelism to Control Theory: eigenvalues in the unitary circle or in the negative half-plane).
I am trying to find something similar for graphs.