# The Fundamental Morphism Theorem in the Categories of Graphs

## Document Type

Presentation Abstract

## Presentation Date

10-29-2012

## Abstract

In the usual Category of Graphs, the graphs allow only one edge to be incident to any two vertices, not necessarily distinct. Also the usual graph morphisms must map edges to edges and vertices to vertices while preserving incidence. We refer to these morphisms as strict morphisms. We relax the condition on the graphs allowing any number of edges to be incident to any two vertices, as well as relaxing the condition on graph morphisms by allowing edges to be mapped to vertices, provided that incidence is still preserved. We call this broader graph category The Category of Conceptual Graphs, and define four other graph categories created by combinations of restrictions of the graph morphisms as well as restrictions on the allowed graphs.

In 1927 Emmy Noether proved three important theorems concerning the behavior of morphisms between groups. These three theorems are called Noether Isomorphism Theorems, and they have been found to apply to morphisms for many mathematical structures. The first of the Noether Isomorphism Theorems when generalized is called the Fundamental Morphism Theorem, and the Noether Isomorphism Theorems follow as corollaries to the Fundamental Morphism Theorem. We investigate how the Fundamental Morphism Theorem applies to these categories of graphs.

## Recommended Citation

Plessas, Demitri, "The Fundamental Morphism Theorem in the Categories of Graphs" (2012). *Colloquia of the Department of Mathematical Sciences*. 411.

https://scholarworks.umt.edu/mathcolloquia/411

## Additional Details

Monday, 29 October 2012

3:10 p.m. in Math 103

4:00 p.m. Refreshments in Math Lounge 109