On the unitary Cayley graph of a ring

Dariush Kiani¹ ; Mohsen Molla Haji Aghaei¹

¹Amirkabir University

The electronic journal of combinatorics, Tome 19 (2012) no. 2

Résumé

Let $R$ be a ring with identity. The unitary Cayley graph of a ring $R$, denoted by $G_{R}$, is the graph, whose vertex set is $R$, and in which $\{x,y\}$ is an edge if and only if $x-y$ is a unit of $R$. In this paper we find chromatic, clique and independence number of $G_{R}$, where $R$ is a finite ring. Also, we prove that if $G_{R} \simeq G_{S}$, then $G_{R/J_{R}} \simeq G_{S/J_{S}}$, where $\rm J_{R}$ and $\rm J_{S}$ are Jacobson radicals of $R$ and $S$, respectively. Moreover, we prove if $G_{R} \simeq G_{M_{n}(F)}$ then $R\simeq M_{n}(F)$, where $R$ is a ring and $F$ is a finite field. Finally, let $R$ and $S$ be finite commutative rings, we show that if $G_{R} \simeq G_{S}$, then $\rm R/ {J}_{R}\simeq S/J_{S}$.

Détail

DOI : 10.37236/2214

Classification : 05C25, 13A99
Mots-clés : unitary Cayley graph, ring, finite field finite ring, independence number, clique number, chromatic number, Jacobson radicals, vertex set, \(G_R \simeq G_S\), \(G_{R/J_R} \simeq G_{S/J_S}\), \(G_R \simeq G_{M_n(F)}\), \(R/J_R \simeq S/J_S\) \(Rsimeq M_n(F)\)