Group Relationships in Homotopy Graph Category
-Research by Alexis Stahl and Ralph Studer ParseError: KaTeX parse error: Undefined control sequence: \$ at position 1: \̲$̲ -Under guidanc…$ -Funded by the Center for Undergraduate Research in Mathematics and NSF
Overview
This code was built to analyze graph transformations, specifically using the Homotopy Category established in the work of Chih-Scull. In the accompanying paper, we build upon the work of Chih-Scull and prove the existence of various group relationships within the Homotopy Category. The following functions cover some of the central concepts we developed including the Spider Nest, the Pleat, and the factor group Aut(G) mod Null(G).
Functions
exponential_object
A graph is defined as a set of vertices and a set of edges, .
The graph category is finite, undirected graphs, where at most one edge connects any pair of vertices, though a vertex can be looped to itself.
The exponential graph, , is defined by:ParseError: KaTeX parse error: Undefined control sequence: \$ at position 1: \̲$̲ -A vertex in V(G^H)V(G) \to V(H) $ -There is an edge if whenever , then .
A homomorphism maps such that if then .
Given two graphs and , exponential_object
returns the subgraph of homomorphisms within the exponential graph, .
spider_web
Given a graph, a spider pair is defined as follows: Let be homomorphisms. are a spider pair if at most one such that .
A spider web of two graphs, , is defined as follows: Let . V(W(G, H)) = {: is a homomorphism} and E(W(G, H)) = {: and are a spider pair}.
spider_nest
A spider nest of a graph, , is defined as .
pleat
A graph, G, is in its stiff form when there exist no distinct vertices and such that .
We then define the pleat of a graph as follows: is the unique stiff subgraph of such that where can be expressed as a series of folds.
factor_group
An automorphism of a graph, G, is a permutation, , of V(G) such that iff .
The automorphism group, Aut(G), is defined as the group of automorphisms of a graph.
We define Null(G) as follows: .
factor_group
returns the quotient also written .
In the example above, we can see that . We know this is true because
In the above example, we see that . We know this is true because contains all autmorphisms of except any permutation that sends (or vertices to their opposite parities).