4 research outputs found
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 be a ring with unity. The cozero-divisor graph of a ring is an
undirected simple graph whose vertices are the set of all non-zero and non-unit
elements of and two distinct vertices and are adjacent if and only
if and . The reduced cozero-divisor graph of a ring
, is an undirected simple graph whose vertex set is the set of all
nontrivial principal ideals of and two distinct vertices and
are adjacent if and only if and . 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 be a ring with unity. The cozero-divisor graph of a ring , denoted
by , is an undirected simple graph whose vertices are the set of
all non-zero and non-unit elements of , and two distinct vertices and
are adjacent if and only if and . 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 . As applications, we
compute the Wiener index of , when either is the product of
ring of integers modulo 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 of integers
modulo