# “The Categories of Graphs”

## Document Type

Presentation Abstract

## Presentation Date

5-5-2011

## Abstract

The most common category considered in (undirected) graph theory is a category where graphs are defined as having at most one edge incident to any two vertices and at most one loop incident to any vertex. The morphisms are usually described as a pair of functions between the vertex sets and edge sets that respect edge incidence. We will relax these conditions to allow multiple edges to be incident to any two vertices, multiple loops to be incident to any vertex, and morphisms will be allowed to map edges to vertices, but they must still preserve edge incidence. With combinations of the three restrictions and relaxations of the three restrictions we define and study five categories of graphs.

One asks when can an abstract system of objects and their morphisms be represented by a familiar system of sets with structure and their structure preserving functions. We answer this question for the categories of graphs giving a characterization of five categories of graphs and their morphisms. We follow the lead and spirit of F. W. Lawvere's groundbreaking categorical characterization of the Category of Sets and Functions (Proc. Nat. Acad. Sci. U.S.A., 52 (1964), pp.1506-1511) and D. Schlomiuk's characterization of the Category of Topological Spaces and Continuous Functions (Trans. Amer. Math. Soc., 149 (1970), pp.259-278).

In both characterizations of the Category of Sets and Functions and the Category of Topological Spaces and Continuous Functions, a list of elementary (or first order) axioms are provided so that when combined with a second order axiom (there exists "small" products and coproducts) a functor equivalence between the axiomatically defined category and the concrete category is formed. We provide such an elementary theory for the five categories of graphs.

## Recommended Citation

Plessas, Demitri, "“The Categories of Graphs”" (2011). *Colloquia of the Department of Mathematical Sciences*. 374.

https://scholarworks.umt.edu/mathcolloquia/374

## Additional Details

Doctoral Dissertation Defense. Link to the presenter's dissertation.

Dissertation Committee:George McRae, Chair (Mathematical Sciences),

Peter Golubstov (Mathematical Sciences),

Kelly McKinnie (Mathematical Sciences),

Nikolaus Vonessen (Mathematical Sciences),

Joel Henry (Computer Science)

Thursday, May 5, 2011

2:00 pm in Native American Center 105