Clear graph of a ring

期刊论文
发表日期：

2024年4月4日

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Shabir Ahmad Mir,
Cihat Abdioğlu,
Nadeem ur Rehman,
Mohd Nazim,
Muhammed Akkafa,
Ece Yetkin Çelikel

摘要

This research article introduces the concept of the clear graph associated with a ring ${\mathcal {R}}$ $R$ with identity, denoted as $Cr({\mathcal {R}})$ $C r (R)$ . This graph comprises vertices of the form $\{(x,u):$ ${(x, u) :$ x is a unit regular element of R and u is a unit of ${\mathcal {R}}$ $R$ } and two distinct vertices (x, u) and (y, v) are adjacent if and only if either $xy=yx=0$ $x y = y x = 0$ or $uv=vu=1$ $u v = v u = 1$ . This research article also focuses on a specific subgraph of $Cr({\mathcal {R}})$ $C r (R)$ denoted as $Cr_2({\mathcal {R}})$ $C r_{2} (R)$ , which is formed by vertices $\{(x,u) :x$ ${(x, u) : x$ is a nonzero unit regular element of $R \}$ $R}$ . The significance of $Cr_2({\mathcal {R}})$ $C r_{2} (R)$ within the context of $Cr{({\mathcal {R}})}$ $C r (R)$ is explored in the article. Taken $Cr_2({\mathcal {R}})$ $C r_{2} (R)$ into consideration, we found connectedness, regularity, planarity, and outer planarity. Moreover, we characterized the ring ${\mathcal {R}}$ $R$ for which $Cr_2({\mathcal {R}})$ $C r_{2} (R)$ is unicyclic, a tree and a split graph. Finally, we have found genus one of $Cr_2({\mathcal {R}})$ $C r_{2} (R)$ .