# How did Spectral Graph theory emerge?

I was wondering how did Spectral graph theory, this multidisciplinary area between Linear Algebra and Graph theory start?

How did it emerge?

Was there a certain problem (maybe graph isomorphism problem?) that evoked its formation?

• Perhaps math history tag is appropriate here? – Yanko Sep 21 '18 at 13:34
• Once the idea of writing down the adjacency matrix is born, it isn't at all hard to use linear algebra to analyse the property of this matrix (and others). Also, the problem of random walk on (possibly directed) graphs are already out there and there too you have all these tools already being used. In a sense, it is just a matter of pulling these together and give it a coherent theory. – user10354138 Sep 21 '18 at 14:02

Well for one thing, certain combinatorial properties of a graph that are of interest to graph theorists and theoretical computer scientists, can be gleaned from analysing the adjacency matrix of the graph. For example, (put informally) a $$k$$-regular graph $$G$$ (where $$k \ge 3$$) is an expander (i.e., lots of edges between $$S$$ and $$V(G) \setminus S$$ for arbitrary nonempty subsets $$S$$ of $$V(G)$$) iff every eigenvalue except the largest is less than $$(1-\epsilon)k$$.