On the Unitary Cayley Graph of a Ring
Keyword(s):
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 $\rm R/ {J}_{R}\simeq S/J_{S}$.
2014 ◽
Vol 13
(05)
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pp. 1350152
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2013 ◽
Vol 12
(04)
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pp. 1250199
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2012 ◽
Vol 12
(03)
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pp. 1250179
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2019 ◽
Vol 19
(09)
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pp. 2050173
2014 ◽
Vol 14
(01)
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pp. 1550002
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Keyword(s):
2018 ◽
Vol 17
(07)
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pp. 1850121
Keyword(s):
2018 ◽
Vol 17
(09)
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pp. 1850178
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