Q:

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? 

A:

The oldest work on spectral graph theory that I am aware of is the work by chemists on Hueckel molecular orbital theory (https://en.wikipedia.org/wiki/Hückel_method). One principal goal of this work was to relate chemical properties of aromatic hydrocarbons to spectral properties of the underlying graphs.
The work of Coulson was very influential.
Chemists also studied the matching polynomial before graph theorists became interested in the topic. (I view 
the matching polynomial as a topic in spectral graph theory.)
Another starting point would be the original paper by Hoffman and Singleton on Moore graphs. It seems hard to overestimate the significance of this work. One thing it did was establish the connection between problems in extremal graph theory and graph spectra.
Certainly there was always the hope that one might use spectral information for graph isomorphism but it does not appear that lead to the development of spectral graph theory.

