What are Simplex Graphs in Graph Theory? [Graph Theory]

What are simplex graphs? This video explains what simplex graphs are in graph theory and shows you how to find the simplex graph of any undirected graph G.

Simplex graphs are based upon the idea of cliques, which are maximally-connected subgraphs of an undirected graph G. In other words, a clique is a subset of the vertices of a graph G, such that each vertex in that subset is adjacent/connected to all other vertices in the subset.

The simplex of a graph is itself a graph with one vertex for each clique in its original graph, with vertices connected to each other according to a special rule covered in the video. If we consider the simplex as a unary operation on graphs, we can also think about the iteration of this operation. That is, taking the simplex of a simplex graph. Some interesting sequences arise as you do this and count the number of vertices or edges in the n-th simplex of a graph G.

Some more fun facts: -Simplex graphs are bipartite (and thus 2-colorable) -Simplex graphs are triangle-free -All N-th simplex graphs, for N greater than 3 have girth 4 -The simplex graph of a cycle graph is a gear graph

Here are some links to check out, including one about a slightly-differently-defined simplex operator:

https://en.wikipedia.org/wiki/Simplex_graph

http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1072/c/dmgt.1072.pdf

If you liked this video, you'd probably like my graph theory playlist: https://www.youtube.com/playlist?list=PLZ2xtht8y2-Jx8hxFvnFQfEej1PzqFbVX

Thanks for watching!

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