The Swiss mathematician Euler is universally credited with having produced the first paper on graph theory. By examining the node-vertex relationships in a graph representing the seven bridges, two islands, and two banks of the Prengel River, he solved the "Koenigsberg Bridge Puzzle" which posed the problem of whether it is possible to begin at any of the four land areas, walk across each bridge exactly once, and return to the starting point.
Euler's work raised the curtain on over two hundred years of mathematical fun-and-games with graphs. The literature is extensive and fascinating. Professor F. Harary prefaces the bibliography of his recent book "Graph Theory" with an appropriate quotation from Lewis Carroll: "and thick and fast they came at last, and more, and more, and more." There are literally dozens of named topological theorems involving intersections, combinations, factorizations, and enumeration of graphs. The theory is rich and the applications are varied. Disregarding the puzzle-oriented papers such as "Calculation of the Maximum Number of Snakes in a Box," a partial and random listing of titles from Harary's extensive bibliography is an indication of the breadth of the applications:
"Valence-Isomerism of Cyclopolyenes"
"A Graph Theory Approach to a Communications Problem"
"Facilities Planning with Graph Theory"
"Space-Time Approach to Quantum Electrodynamics"
"Theory of Games and Economic Behavior"