Line graphs associated to the maximal graph
Let $ R $ be a commutative ring with identity. Let $ G (R) $ denote the maximal graph
associated to $ R $, ie, $ G (R) $ is a graph with vertices as the elements of $ R $, where two
distinct vertices $ a $ and $ b $ are adjacent if and only if there is a maximal ideal of $ R $
containing both. Let $\Gamma (R) $ denote the restriction of $ G (R) $ to non-unit elements
of $ R $. In this paper we study the various graphical properties of the line graph associated
to $\Gamma (R) $, denoted by $(\Gamma (R)) $ such that diameter, completeness, and …
associated to $ R $, ie, $ G (R) $ is a graph with vertices as the elements of $ R $, where two
distinct vertices $ a $ and $ b $ are adjacent if and only if there is a maximal ideal of $ R $
containing both. Let $\Gamma (R) $ denote the restriction of $ G (R) $ to non-unit elements
of $ R $. In this paper we study the various graphical properties of the line graph associated
to $\Gamma (R) $, denoted by $(\Gamma (R)) $ such that diameter, completeness, and …
Let be a commutative ring with identity. Let denote the maximal graph associated to , i.e., is a graph with vertices as the elements of , where two distinct vertices and are adjacent if and only if there is a maximal ideal of containing both. Let denote the restriction of to non-unit elements of . In this paper we study the various graphical properties of the line graph associated to , denoted by such that diameter, completeness, and Eulerian property. A complete characterization of rings is given for which or or . We have shown that the complement of the maximal graph , i.e., the comaximal graph is a Euler graph if and only if has odd cardinality. We also discuss the Eulerian property of the line graph associated to the comaximal graph.
jart.guilan.ac.ir
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