Line cozero-divisor graphs

Let R be a commutative ring. The cozero-divisor graph of R denoted by Γ′(R) is a graph with the vertex set W∗(R), where W∗(R) is the set of all non-zero and non-unit elements of R, and two distinct vertices x and y are adjacent if and only if x ∈/ Ry and y ∈/ Rx. In this paper, we investigate when the cozero-divisor graph is a line graph. We completely present all commutative rings which their cozero-divisor graphs are line graphs. Also, we study when the cozero-divisor graph is the complement of a line graph

Characterization of rings with genus two cozero-divisor graphs

Let $R$ be a ring with unity. The cozero-divisor graph of a ring $R$ is an undirected simple graph whose vertices are the set of all non-zero and non-unit elements of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $x \notin Ry$ and $y \notin Rx$. The reduced cozero-divisor graph of a ring $R$, is an undirected simple graph whose vertex set is the set of all nontrivial principal ideals of $R$ and two distinct vertices $(a)$ and $(b)$ are adjacent if and only if $(a) \not\subset (b)$ and $(b) \not\subset (a)$. In this paper, we characterize all classes of finite non-local commutative rings for which the cozero-divisor graph and reduced cozero-divisor graph is of genus two.Comment: 16 Figure

Wiener index of the Cozero-divisor graph of a finite commutative ring

Let $R$ be a ring with unity. The cozero-divisor graph of a ring $R$, denoted by $\Gamma'(R)$, is an undirected simple graph whose vertices are the set of all non-zero and non-unit elements of $R$, and two distinct vertices $x$ and $y$ are adjacent if and only if $x \notin Ry$ and $y \notin Rx$. In this article, we extend some of the results of [24] to an arbitrary ring. In this connection, we derive a closed-form formula of the Wiener index of the cozero-divisor graph of a finite commutative ring $R$. As applications, we compute the Wiener index of $\Gamma'(R)$, when either $R$ is the product of ring of integers modulo $n$ or a reduced ring. At the final part of this paper, we provide a SageMath code to compute the Wiener index of the cozero-divisor graph of these class of rings including the ring $\mathbb{Z}_{n}$ of integers modulo $n$