Abstract: 
This development provides a formalization of directed graphs, supporting (labelled) multiedges and infinite graphs. A polymorphic edge type allows edges to be treated as pairs of vertices, if multiedges are not required. Formalized properties are i.a. walks (and related concepts), connectedness and subgraphs and basic properties of isomorphisms.
This formalization is used to prove characterizations of Euler Trails, Shortest Paths and Kuratowski subgraphs. 
BibTeX: 
@article{Graph_TheoryAFP,
author = {Lars Noschinski},
title = {Graph Theory},
journal = {Archive of Formal Proofs},
month = apr,
year = 2013,
note = {\url{http://isaafp.org/entries/Graph_Theory.shtml},
Formal proof development},
ISSN = {2150914x},
}
