## Sophia Sage Elia Sciences

### Regular Embeddings of Complete Graphs

Sophia is studying regular embeddings of complete graphs on powers of two vertices. A complete graph is one in which each vertex is connected to each other vertex. Loosely, if one starts with a prime power number of vertices, it is possible to symmetrically connect the vertices in such a way that none of the connecting lines cross on the surface of a torus (think doughnut) with a certain number of holes. There isn’t a constructive way to create a visual representation of the embedding of the graph. Sophia’s project illustrates a fully regular embedding of the complete graph on 8 vertices and locally regular embeddings of the complete graphs on 8, 16, and 32 vertices. She is currently working towards a proof of full regularity for the embedding on 8 vertices through computation of the cartographic and automorphism groups. Embeddings of graphs are connected to a wide range of mathematics. Regular embeddings of complete graphs carry ties to the absolute Galois group, the study of which is an active, exciting field.

Her website is also available here: chromaticmath.com

**Major:**Physics & Mathematics

**Mentor:**Mentor: Bob Jacobsen, Physics