# Colin Defant

## Unitary Cayley Graphs

#### When/Where:

February 14, 2017, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

If $$R$$ is a commutative ring with unity, then the unitary Cayley graph of $$R$$, denoted $$G_R$$ , is the graph with vertex set $$V(G_R)=R$$ and edge set $$E(G_R)=\{\{x,y\} : x-y\text{ is a unit in }R\}$$. We will focus specifically on the unitary Cayley graph of $$\mathbb{Z}/n\mathbb{Z}$$, which we may view as the graph with vertices $$0,1,…,n-1$$ in which two vertices are adjacent if and only if their difference is relatively prime to $$n$$. We provide the values of many graph parameters of these unitary Cayley graphs and find that they are intimately related to some interesting arithmetic functions. We also discuss an open problem concerning the domination numbers of these graphs.