ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P3.05 / 443–458
https://doi.org/10.26493/1855-3974.2645.8fc
(Also available at http://amc-journal.eu)
On the essential annihilating-ideal graph
of commutative rings*
Mohd Nazim , Nadeem ur Rehman †
Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India
Received 30 May 2021, accepted 8 September 2021, published online 9 June 2022
Abstract
Let R be a commutative ring with unity, A(R) be the set of annihilating-ideals of R and
A∗(R) = A(R) \ {0}. In this paper, we introduced and studied the essential annihilating-
ideal graph of R, denoted by EG(R), with vertex set A∗(R) and two distinct vertices I1 and
I2 are adjacent if and only if Ann(I1I2) is an essential ideal of R. We prove that EG(R) is
a connected graph with diameter at most three and girth at most four if EG(R) contains a
cycle. Furthermore, the rings R are characterized for which EG(R) is a star or a complete
graph. Finally, we classify all the Artinian rings R for which EG(R) is isomorphic to some
well-known graphs.
Keywords: Annihilating-ideal graph, zero-divisor graph, complete graph, planar graph, genus of a
graph.
Math. Subj. Class. (2020): 13A15, 05C10, 05C12, 05C25
1 Introduction
Throughout this paper all rings are commutative rings (not a field) with unit element such
that 1 ̸= 0. For a commutative ring R, we use I(R) to denote the set of ideals of R and
I∗(R) = I(R) \ {0}. An ideal I of R is said to be non-trivial if it is nonzero and proper
both. An ideal I of R is said to be annihilator ideal if there is a nonzero ideal J of R such
that IJ = 0. For X ⊆ R, we define annihilator of X as Ann(X) = {r ∈ R : rX = 0}.
We use A(R) to denote the set of annihilator ideas of R and A∗(R) = A(R) \ {0}. We
denote the set of zero-divisors, the set of nilpotent elements, the set of maximal ideals, the
set of minimal prime ideals, and the set of Jacobson radical of a ring R by Z(R), Nil(R),
*The authors are greatly indebted to the referee for his/her constructive comments and suggestion, which
improves the quality of the paper a lot.
†Corresponding author.
E-mail addresses: mnazim1882@gmail.com (Mohd Nazim), nu.rehman.mm@amu.ac.in (Nadeem ur
Rehman)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
444 Ars Math. Contemp. 22 (2022) #P3.05 / 443–458
Max (R), Min(R) and J(R), respectively. A nonzero ideal I of R is called essential,
denoted by I ≤e R, if I has a nonzero intersection with every nonzero ideal of R. Also,
if I is not an essential ideal of R then, it is denoted by I ̸≤e R. A ring R is said to be
reduced, if it has no nonzero nilpotent element. For a nonzero nilpotent element x of R,
we use η to denote the index of nilpotency of x. If S is any subset of R, then S∗ denote the
set S \ {0}. For any undefined notation or terminology in ring theory, we refer the reader
to see [9].
Let G be a graph with vertex set V (G). The distance between two vertices u and v
of G denoted by d(u, v), is the smallest path from u to v. If there is no such path, then
d(u, v) = ∞. The diameter of G is defined as diam(G) = sup{d(u, v) : u, v ∈ V (G)}.
A cycle is a closed path in G. The girth of G denoted by gr(G) is the length of a shortest
cycle in G (gr(G) = ∞ if G contains no cycle). A graph is said to be complete if all its
vertices are adjacent to each other. A complete graph with n vertices is denoted by Kn. If
G is a graph such that the vertices of G can be partitioned into two nonempty disjoint sets
U1 and U2 such that vertices u and v are adjacent if and only if u ∈ U1 and v ∈ U2, then
G is called a complete bipartite graph. A complete bipartite graph with disjoint vertex sets
of size m and n, respectively, is denoted by Km,n. We write Kn,∞ (respectively, K∞,∞)
if one (respectively, both) of the disjoint vertex sets is infinite. A complete bipartite graph
of the form K1,n is called a star graph. A graph G is said to be planar if it can be drawn in
the plane so that its edges intersect only at their ends. A subdivision of a graph is a graph
obtained from it by replacing edges with pairwise internally-disjoint paths. A remarkably
simple characterization of planar graphs was given by Kuratowski in 1930. Kuratowski’s
Theorem says that a graph G is planar if and only if it contains no subdivision of K5 or
K3,3. The genus of a graph G, denoted by γ(G), is the minimum integer k such that the
graph can be drawn without crossing itself on a sphere with k handles (i.e. an oriented
surface of genus k). Thus, a planar graph has genus 0, because it can be drawn on a sphere
without self-crossing. For more details on graph theory, we refer to reader to see [21, 22].
The concept of zero-divisor graph of a commutative ring R, denoted by Γ(R), was
introduced by I. Beck [10]. The vertex set of Γ(R) is Z∗(R) = Z(R) \ {0} (set of nonzero
zero-divisors of R) and two distinct vertices x and y are adjacent if and only if xy =
0, for details see [5, 8, 7]. In [14], Dolžan and Oblak also obtained several interesting
results related with zero-divisor graph of rings and semirings. The zero-divisor graph of a
noncommutative ring has been introduced and studied by Redmond [18], whereas the same
concept for semigroup by Demeyer et al. [13].
In [11], Behboodi et al. generalized the zero-divisor graph to ideals by defining the
annihilating-ideal graph AG(R), with vertex set is A∗(R) and two distinct vertices I1 and
I2 are adjacent if and only if I1I2 = 0. For more details on annihilating-ideal graph, we
refer the reader to see [1, 2, 3, 4, 6, 12, 16].
In [17], M. Nikmehr et al. introduced the essential graph EG(R) with vertex set
Z∗(R) = Z(R)\{0} and two distinct vertices x and y are adjacent if and only if annR(xy)
is an essential ideal of R.
Motivated by [17], we define the essential annihilating-ideal graph of R denoted by
EG(R) with vertex set A∗(R) and two distinct vertices I1 and I2 adjacent if and only if
Ann(I1I2) is an essential ideal of R. In this paper we first prove that AG(R) is a sub-
graph of EG(R) and then studied some basic properties of EG(R) such as connectedness,
diameter, girth and shows that EG(R) is a connected graph with diam(EG(R)) ≤ 3 and
gr(EG(R)) ≤ 4, if EG(R) contains a cycle. In the third section, we determine some condi-
M. Nazim and N. Rehman: On the essential annihilating-ideal graph of commutative rings 445
tions on R under which EG(R) is a star graph or a complete graph. In the last, we identify
all the Artinian rings R for which EG(R) is isomorphic to some well-known graphs.
2 Basic properties of essential annihilating-ideal graph
We begin this section with the following lemma given by [17].
Lemma 2.1 ([17, Lemma 2.1]). Let R be a commutative ring and I be an ideal of R. Then
(1) I +Ann(I) is an essential ideal of R.
(2) If I2 = (0), then Ann(I) is an essential ideal of R.
(3) If R contains no proper essential ideals, then J(R) = (0).
The following lemma is analogue of [17, Lemma 2.2].
Lemma 2.2. Let R be a commutative ring. Then
(1) If I1 and I2 are adjacent in AG(R), then I1 and I2 are also adjacent in EG(R).
(2) If I2 = 0 for some I ∈ A∗(R), then I is adjacent to every other vertex in EG(R).
Proof. (1) Suppose I1 and I2 are adjacent in AG(R), then I1I2 = 0 and so
Ann(I1I2) = R, is an essential ideal of R. Thus I1 and I2 are also adjacent in EG(R).
(2) Suppose that I2 = 0 for some I ∈ A∗(R). Then by Lemma 2.1(2), Ann(I) is an
essential ideal of R. Since Ann(I) ⊆ Ann(IJ) for every J ∈ A∗(R), therefore Ann(IJ)
is also an essential ideal of R. Thus I is adjacent to every other vertex of EG(R).
Let R be a commutative ring. By [11, Theorem 2.1], the annihilating ideal graph
AG(R) is a connected graph with diam(AG(R)) ≤ 3. Moreover, if AG(R) contains
a cycle, then gr(AG(R)) ≤ 4.
In view of part (1) of Lemma 2.2, we have the following result.
Theorem 2.3. Let R be a commutative ring. Then EG(R) is connected with diam(EG(R)) ≤
3. Moreover, if EG(R) contain a cycle, then gr(EG(R)) ≤ 4.
In Lemma 2.2(1), we proved that AG(R) is a spanning subgraph of EG(R) but this
containment may be proper. The following examples shows that AG(R) and EG(R) are
not identical.
Example 2.4.
1. If R = Z16, then AG(R) is P3 and EG(R) is K3.
2. If R = Zp5 , where p is a prime number. Then AG(R) is the following graph and
EG(R) is K4.
Theorem 2.5. Let R be a commutative reduced ring. Then EG(R) = AG(R).
Proof. Clearly, AG(R) ⊆ EG(R). We just have to prove that EG(R) is a subgraph of
AG(R). Suppose on contrary that I1 ∼ I2 is an edge of EG(R) such that I1I2 ̸= 0. Since
R is a reduced ring, then I1I2 ∩ Ann(I1I2) = 0, which implies that Ann(I1I2) is not an
essential ideal of R, a contradiction. Thus I1I2 = 0 and EG(R) = AG(R).
446 Ars Math. Contemp. 22 (2022) #P3.05 / 443–458
Figure 1: The graph AG(Zp5).
Theorem 2.6 ([12, Theorem 1.9(3)]). Let R be a commutative ring with finitely many
minimal primes. Then diam(AG(R)) = 2 if and only if either R is reduced with exactly
two minimal primes and at least three nonzero annihilating-ideals, or R is not reduced,
Z(R) is an ideal whose square is not (0) and for each pair of annihilating-ideals I1 and
I2, I1 + I2 is an annihilating-ideal.
Theorem 2.7. Let R be a commutative ring with |Min(R)| < ∞. Then
(1) If R is reduced ring, then diam(EG(R)) = 2 if and only if |Min(R)| = 2 and R has
at least three nonzero annihilating-ideals. Moreover, in this case
gr(EG(R)) ∈ {4,∞}.
(2) If R is non-reduced, then diam(EG(R)) ≤ 2. Moreover, in this case
gr(EG(R)) ∈ {3,∞}.
Proof. (1) First part is clear from Theorems 2.5 and 2.6. Now, let Min(R) = {P1, P2},
then EG(R) is a complete bipartite graph with partitions V1 = {I ∈ V (EG) : I ⊆ P1} and
V2 = {I ∈ V (EG) : I ⊆ P2} by [12, Theorem 1.2]. Hence gr(EG(R)) ∈ {4,∞}.
(2) Since R is a non-reduced ring, then there is I1 ∈ A∗(R) such that I21 = 0. Thus by
Lemma 2.2(2), I1 is adjacent to every other vertex of EG(R). Hence diam(EG(R)) ≤ 2.
Also, if there are I, J ∈ V (EG(R)) \ {I1} such that I ∼ J is an edge of EG(R),
then I1 ∼ I ∼ J ∼ I1 is a triangle in EG(R). Thus, gr(EG(R)) = 3, otherwise
gr(EG(R)) = ∞.
3 Completeness of essential annihilating-ideal graph
In this section, we characterize commutative rings R for which EG(R) is a star graph or a
complete graph. We begin with the following lemma.
Lemma 3.1. Let R be a commutative nonreduced ring. Then
(1) For every nilpotent ideal I1 of R, I1 is adjacent to every other vertex of EG(R).
(2) The subgraph induced by the nilpotent ideals of R is a complete subgraph of EG(R).
Proof. (1) Suppose that I1 be any nilpotent ideal of R. Let I2 ∈ A∗(R). We show
that Ann(I1I2) ≤e R. Since Ann(I1) ⊆ Ann(I1I2), then it is enough to show that
Ann(I1) ≤e R. Suppose on contrary that Ann(I1) ̸≤e R, then there exists I3 ∈ I∗(R)
such that Ann(I1) ∩ I3 = 0, which implies that rI1 ̸= 0 for every r ∈ I∗3 . Since
0 ̸= rI1 ⊆ I3, then I1 · rI1 = rI21 ̸= 0. Continuing this process, we get rIn1 ̸= 0,
for every positive integer n, which is a contradiction. This complete the proof.
(2) It is clear from (1).
M. Nazim and N. Rehman: On the essential annihilating-ideal graph of commutative rings 447
Lemma 3.2. Let (R,m) be a commutative Artinian local ring. Then EG(R) is a complete
graph.
Proof. Follows from Lemma 3.1.
Lemma 3.3. Let R be a commutative decomposable ring. Then EG(R) is a star graph if
and only if R = F ×D, where F is a field and D is an integral domain.
Proof. (⇒) Suppose that EG(R) is a star graph and let R = R1 × R2, where R1 and R2
are commutative rings. If R1 and R2 both are not fields and I1 ∈ I∗(R1), I2 ∈ I∗(R2),
then (R1 × (0)) ∼ ((0) × R2) ∼ (I1 × (0)) ∼ ((0) × I2) ∼ (R1 × (0)) is a cycle of
length 4 in EG(R), a contradiction. Thus, without loss of generality we can assume that
R1 is a field. We claim that R2 is an integral domain. Suppose on contrary that R2 is not
an integral domain, then there exists I3, I4 ∈ I∗(R1) such that I3I4 = 0. If I3 ̸= I4, then
(R1× (0)) ∼ ((0)×I3) ∼ ((0)×I4) ∼ (R1× (0)) is a triangle in EG(R), a contradiction.
Also, if I3 = I4, then by Lemma 3.1, (R1× (0)) ∼ ((0)× I3) ∼ ((0)×R2) ∼ (R1× (0))
is a triangle in EG(R), again a contradiction. This complete the proof.
(⇐) is clear.
Theorem 3.4. Let R be an Artinian commutative ring with atleast two non-trivial ideals.
Then EG(R) is a star graph if and only if EG(R) ∼= K2.
Proof. (⇒) Suppose EG(R) is a star graph. If R is a local ring, then from Lemma 3.2,
EG(R) is a complete graph. Since EG(R) is a star graph, therefore EG(R) ∼= K2. If R is
non-local ring, then it is decomposable. Thus by Lemma 3.3, R = F × D, where F is a
field and D is an integral domain. Since R is Artinian ring, then D is Artinian and hence
is a field. Thus EG(R) ∼= K2.
(⇐) is evident.
Theorem 3.5. Let R be a commutative ring with at least two non-trivial ideals. Then
EG(R) is a star graph if and only if one of the following holds:
(1) R has exactly two non-trivial ideals.
(2) R = F ×D, where F is a field and D is an integral domain which is not a field.
(3) R has a minimal ideal I1 such that I1 is not an essential ideal of R, I21 = 0 and for
any nonzero annihilating ideal I2 of R, Ann(I2) = I1.
Proof. (⇒) Suppose EG(R) is a star graph. If |A∗(R)| < ∞, then from [11, Theorem 1.1],
R is an Artinian ring. Thus, by Theorem 3.4, EG(R) ∼= K2 and hence (1) hold.
Now, let |A∗(R)| = ∞ and I1 is adjacent to every other vertex of EG(R). We show that I1
is minimal ideal of R. Suppose on contrary that there exists I2 ∈ I∗(R) such that I2 ⊂ I1.
Let I3 ∈ A∗(R) \ {I1, I2}, then Ann(I1I3) ≤e R. Since I2I3 ⊆ I1I3, then Ann(I2I3)
is also essential ideal of R. This implies that I2 is also adjacent to every other vertex of
EG(R), a contradiction. Now, following two cases occur:
Case I: I21 ̸= 0. Then I21 = I1, thus by Brauer’s Lemma [15, p. 172, Lemma 10.22], R
is decomposable. Since |A∗(R)| = ∞ and EG(R) is a star graph. Then from Lemma 3.3,
R = F ×D, where F is a field and D is an integral domain which is not a field. Hence (2)
hold.
Case II: I21 = 0. Let I2 ∈ A∗(R) \ {I1}. Then I2 ̸= Ann(I2), otherwise I22 = 0
448 Ars Math. Contemp. 22 (2022) #P3.05 / 443–458
implies that I2 is also adjacent to every other vertex of EG(R), a contradiction. Now, since
I2 ∼ Ann(I2), then Ann(I2) = I1. If I1 is an essential ideal of R, then Ann(I2) is
also an essential ideal of R. This shows that I2 is also adjacent with every other vertex of
EG(R), which is a contradiction to our assumption that EG(R) is a star graph because we
are assuming that I1 is adjacent with every other vertex of EG(R) and I1 ̸= I2. Hence I1
is not an essential ideal of R.
(⇐) If R has exactly two non-trivial ideals, then R is Artinian ring with |A∗(R)| = 2.
Since EG(R) is connected, therefore EG(R) ∼= K2. If R = F ×D, where F is a field and
D is an integral domain which is not a field, then from Lemma 3.3, EG(R) is a star graph.
Now, suppose that R has a minimal ideal I1 such that I1 is not an essential ideal of R,
I21 = 0 and for any nonzero annihilating ideal I2 of R, Ann(I2) = I1. Let I2, I3 ∈
A∗(R) \ {I1} such that I2 ∼ I3 in EG(R). This implies that Ann(I2I3) ≤e R and
Ann(I2) = I1 = Ann(I3). Since Ann(I2) = Ann(I3) is not an essential of R, there
exists a nonzero ideal I4 of R such that Ann(I2) ∩ I4 = Ann(I3) ∩ I4 = 0. This shows
that rI2 ̸= 0 and rI3 ̸= 0 for every r ∈ I∗4 . On the other hand, since Ann(I2I3) ≤e R, then
Ann(I2I3) ∩ I4 ̸= 0. That is there exists s ∈ I∗4 such that sI2I3 = 0. Now, observe that
sI2 ⊆ I∗4 satisfies sI2 ⊆ Ann(I3), which implies that Ann(I3) ∩ I4 ̸= 0, a contradiction.
This complete the proof.
Theorem 3.6. Let R be a commutative Artinian ring. Then EG(R) is a complete graph if
and only if one of the following holds:
(1) R = F1 × F2, where F1 and F2 are fields.
(2) R is a local ring.
Proof. (⇒) Suppose that EG(R) is a complete graph. Since R is Artinian, then
R ∼= R1 × R2 × · · · × Rn, where Ri is Artinian local ring for each 1 ≤ i ≤ n. The
following cases occur:
Case I: n ≥ 3. Then R1 × (0) × · · · × (0) and R1 × (0) × R3 × · · · × (0) are nonzero
annihilating ideals of R such that (R1 × (0)× · · · × (0)) ̸∼ (R1 × (0)×R3 × · · · × (0))
in EG(R), a contradiction.
Case II: n = 2. We show that R1 and R2 are fields. Suppose on contrary that R1 is not a
field with non-trivial maximal ideal m. Then Ann(((0)× R2) · (m× R2)) = Ann((0)×
R2) = R1 × (0), which is not an essential ideal of R. Thus ((0) × R2) ̸∼ (m × R2) in
EG(R), a contradiction. Hence (2) holds.
Case III: n = 1. Then R is Artinian local ring and (1) holds.
(⇐) If R is local, then from Lemma 3.2, EG(R) is a complete graph. If R = F1 × F2,
where F1 and F2 are fields, then EG(R) ∼= K2.
Theorem 3.7. Let R be a commutative ring with at least one minimal ideal. Then EG(R) ∼=
Km,n, where m,n ≥ 2 if and only if R = D × S, where D and S are integral domains
which are not fields.
Proof. (⇒) Suppose that EG(R) ∼= Km,n, where m,n ≥ 2. Let I1 be minimal ideal of
R. If I21 = 0, then from Lemma 2.2, I1 is adjacent to every other vertex, a contradic-
tion. Thus I21 ̸= 0. Since I1 is minimal, therefore I21 = I1. Therefore, Brauer’s Lemma
[15, p. 172, Lemma 10.22], R = R1×R2, where R1 and R2 are commutative rings. Now,
our objective is to show that R1 and R2 are integral domains. Suppose on contrary that
M. Nazim and N. Rehman: On the essential annihilating-ideal graph of commutative rings 449
R1 is not an integral domain with nonzero annihilating ideal I2. As above, I22 ̸= 0 which
implies that I2 /∈ Ann(I2). Thus (I2×(0)) ∼ ((0)×R2) ∼ (Ann(I2)×(0)) ∼ (I2×(0))
is a triangle in EG(R), a contradiction. Hence R1 is an integral domain. Similarly, one can
prove that R2 is an integral domain. Since m,n ≥ 2, therefore R1 and R2 are not fields.
(⇐) Suppose that R = D × S, where D and S are integral domains which are not fields.
Let U = {I1×(0) : I1 ∈ I∗(D)} and V = {(0)×I2 : I2 ∈ I∗(S)}. Then A∗(R) = U ∪V
such that no two vertices of U or V are adjacent in EG(R). Also, every vertex of U is
adjacent to every vertex of V in EG(R). Thus, EG(R) ∼= Km,n. Since D and S are not
fields, therefore m,n ≥ 2.
Lemma 3.8. Let R be a commutative ring. Then
(1) Let I1, I2, I3 ∈ A∗(R) such that Ann(I1) = Ann(I2). Then I1 ∼ I3 is an edge of
EG(R) if and only if I2 ∼ I3 is an edge of EG(R).
(2) Let I ∈ A∗(R). Then Ann(I) ≤e R if and only if Ann(In) ≤e R for every n ≥ 2.
In particular, if Ann(I3) ≤e R, then Ann(In) ≤e R for every n ≥ 1.
Proof. (1) (⇒) Suppose that I1 ∼ I3 is an edge of EG(R), then Ann(I1I3) ≤e R. We
have to show that Ann(I2I3) ≤e R. Suppose on contrary that Ann(I2I3) is not an essential
ideal of R, then there exits I4 ∈ I∗(R) such that Ann(I2I3) ∩ I4 = 0. This implies that
rI2I3 ̸= 0 for all r ∈ I∗4 . On the other hand, since Ann(I1I3) is an essential ideal of R,
then Ann(I1I3) ∩ I4 ̸= 0. That is there exists some s ∈ I∗4 such that sI1I3 = 0. Now,
observe that sI3 ⊆ I∗4 satisfies sI3 ⊆ Ann(I1) = Ann(I2), which implies that sI2I3 = 0,
a contradiction.
(⇐) Using similar argument as above we get the required result.
(2) (⇒) is clear.
(⇐) Suppose on contrary that Ann(I) is not an essential ideal of R, then there exists
nonzero ideal I1 of R such that Ann(I) ∩ I1 = 0. This implies that rI ̸= 0 for all r ∈ I∗1 .
On the other hand, since Ann(I2) ≤e R, then Ann(I2) ∩ I1 ̸= 0. That is there exists
some s ∈ I∗1 such that sI2 = 0. Now, observe that r = sI ⊆ I∗1 such that rI = 0, a
contradiction.
For the particular case, we need to show that Ann(I2) ≤e R. Suppose on contrary that
there is some I1 ∈ I∗(R) such that Ann(I2) ∩ I1 = 0, which implies that rI2 ̸= 0 for
all r ∈ I∗1 . On the other hand, since Ann(I3) ≤e R, then Ann(I3) ∩ I1 ̸= 0. That is
there exists some s ∈ I∗1 such that sI3 = 0. Now, observe that r = sI2 ⊆ I∗1 such that
rI = 0, which implies that Ann(I) ∩ I1 ̸= 0. Since Ann(I) is a subset of Ann(I2), then
Ann(I2) ∩ I1 ̸= 0, a contradiction.
Theorem 3.9. Let R be a commutative non-reduced ring. Then EG(R) is a complete graph
if and only if Ann(I) ≤e R for every I ∈ A∗(R).
Proof. (⇒) Suppose that EG(R) is a complete graph. We claim that R is indecompos-
able ring. Suppose on contrary that R = R1 × R2, where R1 and R2 are commu-
tative rings. Since R is non-reduced ring, without loss of generality, we can assume
that R1 is non-reduced ring with nonzero nilpotent element x. Let I1 = xR1. Then
Ann((I1 × R2) · ((0) × R2)) = Ann((0) × R2) = R1 × (0), is not an essential ideal
of R, a contradiction to the completeness of EG(R). Let I ∈ A∗(R) be arbitrary. If I is
nilpotent ideal, then from Lemma 3.1(1), Ann(I) ≤e R. Suppose I is not nilpotent ideal.
450 Ars Math. Contemp. 22 (2022) #P3.05 / 443–458
Since R is indecomposable, then I2 ̸= I , which implies that Ann(I3) ≤e R. Hence by
Lemma 3.8(2), Ann(I) ≤e R.
(⇐) is evident.
4 Essential annihilating-ideal graph as some special type of graphs
In this section, we characterize all the Artinian rings R for which EG(R) is a tree, a unicycle
graph, a split graph, a outerplanar graph, a planar graph and a toroidal graph.
Theorem 4.1. Let R be a commutative Artinian ring (not a field). Then EG(R) is a tree
if and only if either R ∼= F1 × F2, where F1 and F2 are fields or R is a local ring with at
most two non-trivial ideals.
Proof. Suppose that EG(R) is a tree. Since R is an Artinain ring, then
R ∼= R1×R2×· · ·×Rn, where each Ri is an Artinian local ring. If n ≥ 3. Consider I1 =
R1×(0)×· · ·×(0), I2 = (0)×R2×(0)×· · ·×(0) and I3 = (0)×(0)×R3×(0)×· · ·×(0).
Then I1 ∼ I2 ∼ I3 ∼ I1 is a cycle of in EG(R), a contradiction.
Suppose n = 2, then we show that R1 and R2 both are fields. Suppose on contrary that
R1 is not a field with nonzero maximal ideal m. Consider J1 = (0) × R2, J2 = m × (0),
J3 = m× R2 and J4 = R1 × (0). Then J1 ∼ J2 ∼ J3 ∼ J4 ∼ J1 is a cycle in EG(R), a
contradiction.
If n = 1, then R is Artinian local ring. Thus by Lemma 3.2, EG(R) is a complete graph.
Since EG(R) is a tree, therefore R has at most two non-trivial ideal.
Converse is clear.
Theorem 4.2. Let R be a commutative Artinian ring (not a field). Then EG(R) is unicycle
if and only if either R ∼= F1 × F2 × F3, where Fi is a field for each 1 ≤ i ≤ 3 or R is an
Artinain local ring with exactly three non-trivial ideals.
Proof. Suppose that EG(R) is unicycle. Since R is Artinian ring, then
R ∼= R1 × R2 × · · · × Rn, where Ri is Artinian local ring for each 1 ≤ i ≤ n. Let
n ≥ 4. Consider I1 = R1 × (0) × · · · × (0), I2 = (0) × R2 × (0) × · · · × (0),
I3 = (0) × (0) × R3 × (0) × · · · × (0) and J1 = (0) × (0) × R3 × (0) × · · · × (0),
J2 = R1 × R2 × (0) × · · · × (0), J3 = (0) × (0) × (0) × R4 × (0) × · · · × (0). Then
I1 ∼ I2 ∼ I3 ∼ I1 as well as J1 ∼ J2 ∼ J3 ∼ J1 are two different cycles in EG(R), a
contradiction. Hence n ≤ 3.
First, let n = 3 and suppose on contrary that R2 is not a field with nonzero maximal ideal
m. Consider I1 = R1 × (0) × (0), I2 = (0) × R2 × (0), I3 = (0) × (0) × R3 and
J1 = R1 × (0)× (0), J2 = (0)×m× (0), J3 = (0)× (0)×R3. Then I1 ∼ I2 ∼ I3 ∼ I1
and J1 ∼ J2 ∼ J3 ∼ J1 are two different cycles in EG(R), a contradiction. Hence Ri is a
field for each 1 ≤ i ≤ 3.
Now, let n = 2. If R1 and R2 both are fields then EG(R) ∼= K2, a contradiction. Thus one
of Ri say R2 is not a field with nonzero maximal ideal m. Then (R1× (0)) ∼ ((0)×m) ∼
((0)×R2) ∼ (R1 × (0)) as well as (R1 ×m ∼ ((0)×m) ∼ ((0)×R2) ∼ (R1 ×m) are
two different cycles in EG(R), again a contradiction.
If n = 1, then R is an Artinian local ring. Thus, by Lemma 3.2, EG(R) is a complete
graph. Since EG(R) is unicycle, R have exactly three non-trivial ideals.
Theorem 4.3 ([21]). Let G be a connected graph. Then G is a split graph if and only if G
contains no induced subgraph isomorphic to 2K2, C4, C5.
M. Nazim and N. Rehman: On the essential annihilating-ideal graph of commutative rings 451
Theorem 4.4. Let R be a commutative Artinian non-local ring. Then EG(R) is split graph
if and only if either R ∼= F1 × F2 × F3 or R ∼= F1 × F2, where Fi is a field for each
1 ≤ i ≤ 3.
Proof. Suppose that EG(R) is a split graph. Since R is Artinian non-local ring, then
R ∼= R1 × R2 × · · · × Rn, where each Ri is an Artinian local ring and n ≥ 2. If n ≥ 4,
then I1 = R1 ×R2 × (0)× · · · × (0) ∼ J1 = (0)× (0)×R3 ×R4 × (0)× · · · × (0) and
I2 = R1 × (0) × R3 × (0) × · · · × (0) ∼ J2 = (0) × R2 × (0) × R4 × (0) × · · · × (0)
induces 2K2 in EG(R), a contradiction. Hence n = 2 or 3. We have following cases:
Case I: If n = 3, then we show that each Ri ia a field. Suppose on contrary that R1 is
not a field with nonzero maximal ideal m. Then (R1 × (0) × (0)) ∼ ((0) × R2 × R3) ∼
(m× (0)× (0)) ∼ ((0)×R2 × (0)) ∼ (R1 × (0)× (0)) is C4 in EG(R), a contradiction.
Hence Ri is a field for each 1 ≤ i ≤ 3.
Case II: Let n = 2 and suppose that R2 is not a field with nonzero maximal ideal m′. Then
(R1 × (0)) ∼ ((0) × R2) ∼ (R1 × m′) ∼ ((0) × m′) ∼ (R1 × (0)) is C4 in EG(R), a
contradiction. Hence R1 and R2 both are fields.
Converse is clear.
Theorem 4.5 ([22]). A graph G is outerplanar if and only if it does not contain a subdivi-
sion of K4 or K2,3.
Theorem 4.6. Let R be a commutative Artinian ring. Then EG(R) is outerplanar if and
only if one of the following holds:
(1) R = F1 × F2 × F3, where Fi is a field for each 1 ≤ i ≤ 3.
(2) R = F1 × F2, where F1 and F2 are fields.
(3) R = F × R1, where F is a field and (R1,m) is a local ring with m is the only
non-trivial ideal of R1.
(4) R is a local ring with at most three non-trivial ideals.
Proof. Suppose that EG(R) is outerplanar. Since R is Artinian ring, then
R ∼= R1×R2×· · ·×Rn, where each Ri is Artinian local ring. If n ≥ 4, then the set {I1 =
R1×(0)×· · ·×(0), I2 = (0)×R2×(0)×· · ·×(0), I3 = (0)×(0)×R3×(0)×· · ·×(0),
I4 = (0)× (0)× (0)×R4× (0)×· · ·× (0)} induces K4 in EG(R), a contradiction. Hence
n ≤ 3. The following cases occur:
Case I: n = 3. We claim that Ri is a field for each 1 ≤ i ≤ 3. Suppose on contrary that R2
is not a field with nonzero maximal ideal m. Then the set
{R1 × (0) × (0), R1 × m × (0), (0) × m × (0), (0) × (0) × R3, (0) × R2 × R3} in-
duces a copy of K2,3 with partition sets A = {(0) × (0) × R3, (0) × R2 × R3} and
B = {R1 × (0) × (0), R1 × m × (0), (0) × m × (0)}, a contradiction. Therefore Ri is a
field for each 1 ≤ i ≤ 3.
Case II: n = 2 and let Ri is not a field with nonzero maximal ideal mi for each i =
1, 2. Then the set {R1 × (0), (0) × R2,m1 × (0), (0) × m2} induces a copy of K4 in
EG(R), a contradiction. Hence one of Ri (say R1) must be a field. Let I be a non-trivial
ideal of R2 other than maximal ideal m2. Then the set {R1 × (0), R1 × m2, (0) × R2,
(0) × m2, (0) × I} induces a copy of K2,3 with partition sets A = {R1 × (0), R1 × m2}
and B = {(0)×R2, (0)×m2, (0)× I} in EG(R), a contradiction. Hence R2 is a field or
452 Ars Math. Contemp. 22 (2022) #P3.05 / 443–458
has unique non-trivial ideal.
Case III: n = 1, then R is an Artinian local ring. Thus by Lemma 3.2, EG(R) is a complete
graph. Since EG(R) is outerplanar, R have at most three non-trivial ideals.
Converse follows from Lemma 3.2, Theorem 4.5, Figures 2 and 3.
F1 × (0)× F3
(0)× F2 × F3
(0)× (0)× F3
F1 × F2 × (0)
F1 × (0)× (0)
(0)× F2 × (0)
Figure 2: The graph EG(F1 × F2 × F3).
0×m F ×m
F × (0) (0)×R
Figure 3: The graph EG(F ×R1), where m is the only non-trivial ideal of R1.
Lemma 4.7 ([20, Proposition 2.7]). If (R,m) is an Artinian local ring and there is an ideal
I of R such that I ̸= mi for every i, then R has at least three distinct non-trivial ideals
J,K and L such that J,K,L ̸= mi for each i.
Theorem 4.8 (Kuratowski’s Theorem). A graph G is planar if and only if it contains no
subdivision of K5 or K3,3.
Lemma 4.9. Let (R,m) be a commutative Artinian local ring. Then EG(R) is planar if
and only if R have at most four non-trivial ideals.
Proof. It is clear from Lemma 3.2 and Theorem 4.8.
Theorem 4.10. Let R be a commutative Artinian ring. Then EG(R) is planar graph if and
only if one of the following hold:
(1) R = F1 × F2 × F3, where Fi is a field for each 1 ≤ i ≤ 3.
M. Nazim and N. Rehman: On the essential annihilating-ideal graph of commutative rings 453
(2) R has at most four non-trivial ideals.
Proof. Suppose that EG(R) is a planar graph. If |A∗(R)| ≤ 4, then (2) holds. Thus, we
assume that |A∗(R)| ≥ 5. Since R is Artinian ring, then R ∼= R1 ×R2 × · · · ×Rn, where
each Ri is Artinian local ring. If n ≥ 4, then the set {R1 × (0) × · · · × (0), R1 × R2 ×
(0)×· · ·× (0), (0)×R2× (0)×· · ·× (0)} ∪ {(0)× (0)×R3×R4× (0)×· · ·× (0), (0)×
(0) × R3 × (0) × · · · × (0), (0) × (0) × (0) × R4 × (0) × · · · × (0)} induces a copy of
K3,3 in EG(R), a contradiction. Hence n ≤ 3. The following cases occur:
Case I: n = 3. We claim that Ri is a field for each 1 ≤ i ≤ 3. Suppose on con-
trary that one of Ri say R2 is not a field with nonzero maximal ideal m. Then the set
{R1×(0)×(0), R1×m×(0), (0)×m×(0), (0)×m×R3, (0)×(0)×R3, (0)×R2×R3}
induces a copy of K3,3 with partition sets A = {R1×(0)×(0), R1×m×(0), (0)×m×(0)}
and B = {(0)×(0)×R3, (0)×m×R3, (0)×R2×R3} in EG(R), a contradiction. Hence,
(1) satisfied.
Case II: n = 2. Since |A∗(R)| ≥ 5, then one of Ri is not a field for some i = 1, 2.
Suppose that R1 is not a field with nonzero maximal ideal m1. If R2 is a field, then
|A∗(R)| ≥ 5 shows that R1 have at least two non-trivial ideals. Let I be a non-trivial ideal
of R1 other than the maximal ideal. Then the set {R1×(0),m1×(0), I×(0)}∪{(0)×R2,
m1 ×R2, I ×R2} induces a copy of K3,3 in EG(R), a contradiction.
Now, if R2 is not a field with nonzero maximal ideal m2, then the set
{R1 × (0), (0) × m2, R1 × m2} ∪ {(0) × R2,m1 × (0),m1 × R2} induces a copy of
K3,3 in EG(R), again a contradiction.
Case III: n = 1. Then R is an Artinian local ring. Thus, by Lemma 3.2, EG(R) is
a complete graph. Since |A∗(R)| ≥ 5, then EG(R) contains a copy of K5, which is a
contradiction.
Conversely, If R is an Artinian ring with at most four non-trivial ideals, then by Theo-
rem 4.8, EG(R) is planar. Also, if R = F1×F2×F3, where Fi is a field for each 1 ≤ i ≤ 3,
then from Figure 2, EG(R) is planar.
Lemma 4.11 ([22]). γ(Kn) = ⌈ 112 (n − 3)(n − 4)⌉, where ⌈x⌉ is the least integer that is
greater than or equal to x. In particular, γ(Kn) = 1 if n = 5, 6, 7.
Lemma 4.12 ([22]). γ(Km,n) = ⌈ 14 (m − 2)(n − 2)⌉, where ⌈x⌉ is the least integer that
is greater than or equal to x. In particular, γ(K4,4) = γ(K3,n) = 1 if n = 3, 4, 5, 6.
Theorem 4.13. Let (R,m) be a commutative Artinian local ring. Then γ(EG(R)) = 1 if
and only if R have at least five and at most seven non-trivial ideals.
Proof. Since (R,m) is an Artinian local ring, then from Lemma 3.2, EG(R) is a complete
graph. Thus, by Lemma 4.11, 5 ≤ r ≤ 7, where r is the number of non-trivial ideals of
R.
Theorem 4.14. Let R be a commutative Artinian ring such that R = F1 × F2 × · · · × Fn,
where n ≥ 4 and Fi is a field for each 1 ≤ i ≤ n. Then γ(EG(R)) = 1 if and only if
n = 4.
Proof. Since R is a reduced ring, EG(R) = AG(R) by Theorem 2.5. Hence the result
follows from [19, Theorem 2].
454 Ars Math. Contemp. 22 (2022) #P3.05 / 443–458
Theorem 4.15. Let R be a commutative Artinian ring such that R = R1×R2×· · ·×Rn,
where n ≥ 2 and each (Ri,mi) is an Artinian local ring with mi ̸= 0. Let ηi be the
nilpotency of mi. Then γ(EG(R)) = 1 if and only if n = 2 and m1 and m2 are the only
non-trivial ideals of R1 and R2 respectively.
Proof. Suppose that γ(EG(R)) = 1. If n ≥ 3, then the set {mη1−11 × (0)×· · ·× (0), (0)×
mη2−12 × (0)×· · ·× (0),m
η1−1
1 ×m
η2−1
2 × (0)×· · ·× (0)}∪{(0)× (0)×R3× (0)×· · ·×
(0), (0)× (0)×m3 × (0)× · · · × (0),m1 ×m2 ×m3 × (0)× · · · × (0),m1 × (0)×m3 ×
(0)× · · · × (0), (0)×m2 ×m3 × (0)× · · · × (0),m1 × (0)×R3 × (0)× · · · × (0), (0)×
m2 × R3 × (0) × · · · × (0)} induces a copy of K3,7 in EG(R). Thus, from Lemma 4.12,
γ(EG(R)) > 1, a contradiction. Hence n = 2.
Suppose I is non-trivial ideal of R1 such that I ̸= m1. Then the set {R1 × (0),
m1 × (0), R1 × m2, I × (0), I × m2} ∪ {(0) × R2, (0) × m2,m1 × R2,m1 × m2} in-
duces a copy of K4,5 in EG(R). By Lemma 4.12, γ(EG(R)) > 1, a contradiction. Hence
R1 has unique non-trivial ideal m1. Similarly, we can show that R2 has unique non-trivial
ideal m2.
Conversely, let R = R1×R2, where m1 and m2 are the only non-trivial ideals of R1 and R2
respectively, then |A∗(R)| = 7. It is easy to see that the set {R1 × (0),
m1 × (0), R1 × m2} ∪ {(0) × R2, (0) × m2,m1 × R2} induces a copy of K3,3, which
implies that K3,3 ≤ EG(R) ≤ K7. Hence, by Lemma 4.11 and 4.12, γ(EG(R)) = 1.
Theorem 4.16 ([19, Theorem 4]). Let R = R1 × R2 × F be a commutative ring, where
each (Ri,mi) is a local ring with mi ̸= 0 and F is a field. Let ηi be the nilpotency of mi.
Then γ(AG(R)) > 1.
Theorem 4.17 ([19, Theorem 5]). Let R = R1 × F1 × F2 × · · · × Fm be a commutative
ring, where each (R1,m1) is a local ring with m1 ̸= 0 and each Fj is a field. Let η1 be the
nilpotency of m1 and m ≥ 3. Then γ(AG(R)) > 1.
Theorem 4.18. Let R be a commutative Artinian ring such that R = R1×R2×· · ·×Rn×
F1 × F2 × · · · × Fm, where each (Ri,mi) is an Artinian local ring with mi ̸= 0 and each
Fj is a field. Let ηi be the nilpotency of mi and n ≥ 2 or m ≥ 3. Then γ(EG(R)) > 1.
Proof. Follows from Theorems 4.16 and 4.17.
Theorem 4.19. Let R be a commutative Artinian ring such that R = R1×F1×F2, where
(R1,m) is an Artinian local ring and F1 and F2 are fields. Let η be the nilpotency of m.
Then γ(EG(R)) = 1 if and only if η = 2 and m is the only non-trivial ideal of R1.
Proof. Suppose that η = 2 and m is the only non-trivial ideal of R1. Then from Figure 5,
we get γ(EG(R)) = 1, where a = m× (0)× (0), b = R1 × (0)× (0), c = m× F1 × F2,
d = (0) × F1 × F2, e = m × (0) × F2, f = (0) × F1 × (0), g = R1 × F1 × (0),
h = R1 × (0)× F2, i = (0)× (0)× F2, j = m× F1 × (0).
Conversely, assume that γ(EG(R)) = 1. Let J be a non-trivial ideal of R1 such that J ̸= m.
Then the set {m× (0)× (0),m× F1 × (0), J × F1 × (0), (0)× F1 × (0)} ∪ {J × (0)×
(0),m× (0)× F2, J × (0)× F2, (0)× (0)× F2, R1 × (0)× (0)} induces a copy of K4,5
in EG(R), which is a contradiction. Hence m is the only non-trivial ideal of R1.
Theorem 4.20. Let R be a commutative Artinian ring such that R = R1 × F , where
(R1,m) is an Artinian local ring and F is a field. Let η be the nilpotency of m. Then
γ(EG(R)) = 1 if and only if one of the following holds:
M. Nazim and N. Rehman: On the essential annihilating-ideal graph of commutative rings 455
m1 ×m2 m1 ×m2
m1 ×m2 m1 ×m2
(0)×m2
m1 ×R2
m1 ×R2
(0)×R2 (0)×R2
R1 ×m2
R1 ×m2
R1 × (0) R1 × (0)
m1 × (0)
Figure 4: Toroidal embedding of EG(R1 × R2), where mi is the only non-trivial ideal of
Ri for i = 1, 2.
a a
aa
b
f f
e e
i
i
g
h
j
j
d c
Figure 5: Toroidal embedding of EG(R1 × F1 × F2), where m is the only non-trivial ideal
of R1.
(1) η = 3 and m and m2 are the only non-trivial ideals of R1.
(2) η = 4 and m, m2 and m3 are the only non-trivial ideals of R1.
Proof. Suppose that γ(EG(R)) = 1. If η ≥ 5, then the set {mη−1 × (0),mη−2 ×
(0),mη−3 × (0)}∪{R1 × (0),m× (0), (0)×F,mη−1 ×F,mη−2 ×F,mη−3 ×F,m×F}
456 Ars Math. Contemp. 22 (2022) #P3.05 / 443–458
m2 × Fm2 × (0)
m× (0)
(0)× F
(0)× F
R1 × (0) R1 × (0)
R1 × (0)R1 × (0)
m× F
Figure 6: Toroidal embedding of EG(R1 × F ), where m and m2 are only non-trivial ideals
of R1.
induces a copy of K3,7. Thus, by Lemma 4.12, γ(EG(R)) > 1, a contradiction. Hence
η ≤ 4. We have following cases:
Case I: η = 2. Let J be a non-trivial ideal of R1 such that J ̸= m. Then by Lemma 4.7,
R1 has at least three non-trivial ideals I1, I2 and I3 such that I1, I2, I3 ̸= m. We can see
that the set {R1×(0), J×(0), I1×(0), I2×(0)}∪{(0)×F, J×F, I1×F, I2×F,m×F}
induces a copy of K3,7 in EG(R), a contradiction. Hence m is the only non-trivial ideal of
R1. It follows from Theorem 4.10 that EG(R) is a planar graph, a contradiction.
Case II: η = 3. Let I be a non-trivial ideal of R1 such that I ̸= m,m2. Then by Lemma 4.7,
R1 has at least three non-trivial ideals I1, I2 and I3 such that I1, I2, I3 ̸= m,m2. It is easy
to see that the set {R1 × (0),m × (0),m2 × (0)} ∪ {I × (0), I1 × (0), I2 × (0), 0 × F,
m × F,m2 × F, I × F} induces a copy of K3,7 in EG(R), a contradiction. Hence m and
m2 are the only non-trivial ideals of R1.
Case III: η = 4. Let I be a non-trivial ideal of R1 such that I ̸= mi for each
i = 1, 2, 3. Then by Lemma 4.7, R1 has at least three non-trivial ideals I1, I2 and I3
such that I1, I2, I3 ̸= mi for each i = 1, 2, 3. It is easy to see that the set {m × (0),
m2 × (0),m3 × (0)} ∪ {R1 × (0), I × (0), I1 × (0), I2 × (0),m×F,m2 × (0),m3 × (0)}
induces a copy of K3,7 in EG(R), a contradiction. Hence m, m2 and m3 are the only non-
trivial ideals of R1.
Conversely, if m and m2 are the only non-trivial ideals of R1, then |A∗(R)| = 6 and the
set {R1 × (0),m× (0),m2 × (0)} ∪ {(0)× F,m× F,m2 × F} induces a copy of K3,3 in
EG(R). Thus, K3,3 ≤ EG(R) ≤ K6, which implies that γ(EG(R)) = 1.
Now, if m, m2 and m3 are the only non-trivial ideals of R1. Then from Figure 7,
γ(EG(R)) = 1.
M. Nazim and N. Rehman: On the essential annihilating-ideal graph of commutative rings 457
m2 × F
m2 × F
m2 × (0)
m2 × (0)
(0)× F (0)× F
m3 × (0)
m× (0) m3 × F
R1 × (0)
R1 × (0) R1 × (0)
R1 × (0)
m× F m× F
Figure 7: Toroidal embedding of EG(R1 × F ), where m, m2 and m3 are non-trivial ideals
of R1.
ORCID iDs
Mohd Nazim https://orcid.org/0000-0001-8817-4336
Nadeem ur Rehman https://orcid.org/0000-0003-3955-7941
References
[1] G. Aalipour, S. Akbari, M. Behboodi, R. Nikandish, M. J. Nikmehr and F. Shaveisi, The clas-
sification of the annihilating-ideal graphs of commutative rings, Algebra Colloq. 21 (2014),
249–256, doi:10.1142/s1005386714000200.
[2] G. Aalipour, S. Akbari, R. Nikandish, M. Nikmehr and F. Shaveisi, On the coloring of the
annihilating-ideal graph of a commutative ring, Discrete Math. 312 (2012), 2620–2626, doi:
10.1016/j.disc.2011.10.020.
[3] G. Aalipour, S. Akbari, R. Nikandish, M. J. Nikmehr and F. Shaveisi, Minimal prime ideals
and cycles in annihilating-ideal graphs, Rocky Mountain J. Math. 43 (2013), 1415–1425, doi:
10.1216/rmj-2013-43-5-1415.
[4] M. Ahrari, S. A. S. Sabet and B. Amini, On the girth of the annihilating-ideal graph of a
commutative ring, J. Linear Topol. Algebra 4 (2015), 209–216.
[5] S. Akbari and A. Mohammadian, On the zero-divisor graph of a commutative ring, J. Algebra
274 (2004), 847–855, doi:10.1016/s0021-8693(03)00435-6.
[6] F. Aliniaeifard, M. Behboodi, E. Mehdi-Nezhad and A. M. Rahimi, On the diameter and girth
of an annihilating-ideal graph, 2014, arXiv:1411.4163v1 [math.CO].
[7] D. D. Anderson and M. Naseer, Beck’s coloring of a commutative ring, J. Algebra 159 (1993),
500–514, doi:10.1006/jabr.1993.1171.
[8] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra
217 (1999), 434–447, doi:10.1006/jabr.1998.7840.
458 Ars Math. Contemp. 22 (2022) #P3.05 / 443–458
[9] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley
Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.
[10] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), 208–226, doi:10.1016/
0021-8693(88)90202-5.
[11] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra
Appl. 10 (2011), 727–739, doi:10.1142/s0219498811004896.
[12] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Algebra
Appl. 10 (2011), 741–753, doi:10.1142/s0219498811004902.
[13] F. R. DeMeyer, T. McKenzie and K. Schneider, The zero-divisor graph of a commutative semi-
group, Semigroup Forum 65 (2002), 206–214, doi:10.1007/s002330010128.
[14] D. Dolžan and P. Oblak, The zero-divisor graphs of rings and semirings, Internat. J. Algebra
Comput. 22 (2012), 1250033, 20, doi:10.1142/s0218196712500336.
[15] T. Y. Lam, A First Course in Non-Commutative Rings, Springer-Verlag, New York, 1991, doi:
10.1007/978-1-4684-0406-7.
[16] M. J. Nikmehr and S. M. Hosseini, More on the annihilator-ideal graph of a commutative ring,
J. Algebra Appl. 18 (2019), 1950160, 14, doi:10.1142/s0219498819501603.
[17] M. J. Nikmehr, R. Nikandish and M. Bakhtyiari, On the essential graph of a commutative ring,
J. Algebra Appl. 16 (2017), 1750132, 14, doi:10.1142/s0219498817501328.
[18] S. P. Redmond, The zero-divisor graph of a non-commutative ring, in: Internat. J. Commutative
Rings, Nova Sci. Publ., Hauppauge, NY, volume 1, 2002.
[19] K. Selvakumar and P. Subbulakshmi, Classification of rings with toroidal annihilating-ideal
graph, Commun. Comb. Optim. 3 (2018), 93–119, doi:10.22049/cco.2018.26060.1072.
[20] K. Selvakumar, P. Subbulakshmi and J. Amjadi, On the genus of the graph associated to
a commutative ring, Discrete Math. Algorithms Appl. 9 (2017), 1750058, 11, doi:10.1142/
s1793830917500586.
[21] D. B. West, Introduction to Graph Theory, Prentice-Hall of India, New Delhi, 2002.
[22] A. T. White, Graphs, Groups and Surfaces, North-Holland Mathematics Studies, No. 8, North-
Holland Publishing Co., Amsterdam, 1973.