5 research outputs found
Characterization of rings with genus two prime ideal sum graphs
Let be a commutative ring with unity. The prime ideal sum graph of the
ring is a simple undirected graph whose vertex set is the set of nonzero
proper ideals of and two distinct vertices and are adjacent if and
only if is a prime ideal of . In this paper, we characterize all the
finite non-local commutative rings whose prime ideal sum graph is of genus .Comment: 13 figures, Asian-European Journal of Mathematics, Accepte
Embedding of prime ideal sum graph of a commutative ring on surfaces
Let be a commutative ring with unity. The prime ideal sum graph
of the ring is the simple undirected graph whose vertex set
is the set of all nonzero proper ideals of and two distinct vertices
and are adjacent if and only if is a prime ideal of . In this
paper, we classify non-local commutative rings such that is
of crosscap at most two. We prove that there does not exist a finite non-local
commutative ring whose prime ideal sum graph is projective planar. Further, we
classify non-local commutative rings of genus one prime ideal sum graphs.
Moreover, we classify finite non-local commutative rings for which the prime
ideal sum graph is split graph, threshold graph, cograph, cactus graph and
unicyclic, respectively
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