**This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.**

### Abstract

This entry presents a general library for undirected graph theory -
enabling reasoning on simple graphs and undirected graphs with loops.
It primarily is inspired by Noschinski's basic ugraph definition
in the Girth Chromatic Entry,
however generalises it in a number of
ways and significantly expands on the range of basic graph theory
definitions formalised. Notably, this library removes the constraint
of vertices being a type synonym with the natural numbers which causes
issues in more complex mathematical reasoning using graphs, such as
the Balog Szemeredi Gowers theorem which this library is used for.
Secondly this library also presents a locale-centric approach,
enabling more concise, flexible, and reusable modelling of different
types of graphs. Using this approach enables easy links to be made
with more expansive formalisations of other combinatorial structures,
such as incidence systems, as well as various types of formal
representations of graphs. Further inspiration is also taken from
Noschinski's Directed Graph library for some proofs and
definitions on walks, paths and cycles, however these are much
simplified using the set based representation of graphs, and also
extended on in this formalisation.