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Search: All articles in the CMB digital archive with keyword Diameter

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1. CMB 2013 (vol 57 pp. 413)

Samei, Karim
On the Comaximal Graph of a Commutative Ring
Let $R$ be a commutative ring with $1$. In [P. K. Sharma, S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra 176(1995) 124-127], Sharma and Bhatwadekar defined a graph on $R$, $\Gamma(R)$, with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra + Rb = R$. In this paper, we consider a subgraph $\Gamma_2(R)$ of $\Gamma(R)$ which consists of non-unit elements. We investigate the behavior of $\Gamma_2(R)$ and $\Gamma_2(R) \setminus \operatorname{J}(R)$, where $\operatorname{J}(R)$ is the Jacobson radical of $R$. We associate the ring properties of $R$, the graph properties of $\Gamma_2(R)$ and the topological properties of $\operatorname{Max}(R)$. Diameter, girth, cycles and dominating sets are investigated and the algebraic and the topological characterizations are given for graphical properties of these graphs.

Keywords:comaximal, Diameter, girth, cycles, dominating set

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