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<p>In this paper, we consider the non-comaximal ideal graph NC(<i>R</i>) of nontrivial ideals of a commutative ring <i>R</i> with unity. We characterize a ring <i>R</i> for which the graph NC(<i>R</i>) is totally disconnected, complete, connected, bipartite, unicyclic, split, cograph, chordal and also provide a necessary and sufficient condition under which the graph NC(<i>R</i>) has a perfect code. We also characterize a ring <i>R</i> whose non-comaximal ideal graph NC(<i>R</i>) is of genus at most one and crosscap at most two. Finally, we study the perfectness of NC(<i>R</i>) and provide a necessary and sufficient condition for the graph NC(<i>R</i>) to be Hamiltonian.</p>

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Some results on non-comaximal ideal graph of a ring

  • 期刊论文

  • 发表日期：

    2025年6月26日

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  • Biswaranjan Khanra,
  • Reza Nikandish

摘要

In this paper, we consider the non-comaximal ideal graph NC(R) of nontrivial ideals of a commutative ring R with unity. We characterize a ring R for which the graph NC(R) is totally disconnected, complete, connected, bipartite, unicyclic, split, cograph, chordal and also provide a necessary and sufficient condition under which the graph NC(R) has a perfect code. We also characterize a ring R whose non-comaximal ideal graph NC(R) is of genus at most one and crosscap at most two. Finally, we study the perfectness of NC(R) and provide a necessary and sufficient condition for the graph NC(R) to be Hamiltonian.