On the genus of the annhilator graph of a commutative ring
Let R be a commutative ring and Z(R)* be its set of non-zero zero-divisors. The annihilator graph of a commutative ring R is the simple undirected graph AG(R) with vertices Z(R)*, and two distinct vertices x and y are adjacent if and only if ann(xy)≠ann(x)∪ann(y). The notion of annihilator graph has...
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Chelvam, T.T. Selvakumar, K. 2019-06-18T18:18:04Z 2019-06-18T18:18:04Z 2017 On the genus of the annhilator graph of a commutative ring / T.T. Chelvam, K. Selvakumar // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 2. — С. 191-208. — Бібліогр.: 26 назв. — англ. 1726-3255 2010 MSC:05C99, 05C15, 13A99. https://nasplib.isofts.kiev.ua/handle/123456789/156637 Let R be a commutative ring and Z(R)* be its set of non-zero zero-divisors. The annihilator graph of a commutative ring R is the simple undirected graph AG(R) with vertices Z(R)*, and two distinct vertices x and y are adjacent if and only if ann(xy)≠ann(x)∪ann(y). The notion of annihilator graph has been introduced and studied by A. Badawi [7]. In this paper, we determine isomorphism classes of finite commutative rings with identity whose AG(R) has genus less or equal to one This work was supported by the UGC-BSR One-time grant and UGC Major Research Project (F. No. 42-8/2013(SR)) of University Grants Commission, Government of India through first and second authors respectively. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On the genus of the annhilator graph of a commutative ring Article published earlier |
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On the genus of the annhilator graph of a commutative ring Chelvam, T.T. Selvakumar, K. |
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On the genus of the annhilator graph of a commutative ring |
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On the genus of the annhilator graph of a commutative ring |
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on the genus of the annhilator graph of a commutative ring |
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Let R be a commutative ring and Z(R)* be its set of non-zero zero-divisors. The annihilator graph of a commutative ring R is the simple undirected graph AG(R) with vertices Z(R)*, and two distinct vertices x and y are adjacent if and only if ann(xy)≠ann(x)∪ann(y). The notion of annihilator graph has been introduced and studied by A. Badawi [7]. In this paper, we determine isomorphism classes of finite commutative rings with identity whose AG(R) has genus less or equal to one
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On the genus of the annhilator graph of a commutative ring / T.T. Chelvam, K. Selvakumar // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 2. — С. 191-208. — Бібліогр.: 26 назв. — англ. |
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“adm-n4” 22:47 page #13
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 24 (2017). Number 2, pp. 191–208
© Journal “Algebra and Discrete Mathematics”
On the genus of the annihilator graph
of a commutative ring∗
T. Tamizh Chelvam and K. Selvakumar
Communicated by V. V. Kirichenko
Abstract. Let R be a commutative ring and Z(R)∗ be its
set of non-zero zero-divisors. The annihilator graph of a commu-
tative ring R is the simple undirected graph AG(R) with vertices
Z(R)∗, and two distinct vertices x and y are adjacent if and only
if ann(xy) 6= ann(x) ∪ ann(y). The notion of annihilator graph has
been introduced and studied by A. Badawi [7]. In this paper, we
determine isomorphism classes of finite commutative rings with
identity whose AG(R) has genus less or equal to one.
1. Introduction
The study of algebraic structures, using the properties of graphs,
became an exciting research topic in the past twenty years, leading to
many fascinating results and questions. I. Beck[8] began the study of
associating a graph called the zero-divisor graph Γ0(R) to a commutative
ring R and was mainly interested in the coloring of the zero-divisor graph.
For a commutative ring R, the zero-divisor graph is the simple graph with
R as the vertex set and two distinct elements x and y are adjacent if and
only if xy = 0 [8]. D. F. Anderson and P. S. Livingston[3] modified and
∗This work was supported by the UGC-BSR One-time grant and UGC Major
Research Project (F. No. 42-8/2013(SR)) of University Grants Commission, Government
of India through first and second authors respectively.
2010 MSC: 05C99, 05C15, 13A99.
Key words and phrases: commutative ring, annihilator graph, genus, planar,
local rings.
“adm-n4” 22:47 page #14
192Genus of the annihilator graph of a commutative ring
studied the zero-divisor graph Γ(R) as the graph with vertex set as the
nonzero zero-divisors Z(R)∗ of R and two distinct elements x, y ∈ Z∗(R)
are adjacent if and only if xy = 0. Thereafter, several graphs have been
associated with commutative rings. These graphs exhibit the interplay
between the algebraic properties of R and graph theoretical properties of
the associated graph. For a ∈ R, let ann(a) = {d ∈ R : da = 0} be the
annihilator of a ∈ R. In 2014, A. Badawi [7] introduced the annihilator
graph AG(R) as the simple graph with vertex set Z(R)∗, and two distinct
vertices x and y are adjacent if and only if ann(xy) 6= ann(x)∪ann(y). One
can see that the zero-divisor graph Γ(R) is a subgraph of the annihilator
graph AG(R).
The main objective of topological graph theory is to embed a graph
into a surface. Let Sk denote the sphere with k handles, where k is a
nonnegative integer, that is, Sk is an oriented surface of genus k. The
genus of a graph G, denoted g(G), is the minimal integer n such that the
graph can be embedded in Sn. Intuitively, G is embedded in a surface if
it can be drawn in the surface so that its edges intersect only at their
common vertices. A graph G with genus 0 is called a planar graph, whereas
a graph G with genus 1 is called a toroidal graph. Further note that if H
is a subgraph of a graph G, then g(H) 6 g(G). For details on the notion
of embedding a graph in a surface, see [26].
Many research articles have appeared on the genus of zero divisor
graphs of commutative rings. In particular, there are many papers [1,2,9,
15,24], where the planarity of zero-divisor graphs has been discussed. The
question addressed in these papers is this:For which finite commutative
rings R is Γ(R) planar? A partial answer was given in [1], but the question
remained open for local rings of order 32. In [15], and independently in
both [9] and [24], it was shown that no local ring of order 32 has the
planar zero divisor graph. Furthermore, Smith [15] gave a complete list
of finite planar rings; this list included the 2 infinite families Z2 × F
and Z3 × F , where F is any finite field, and the 42 other isomorphism
classes of finite planar rings. H.J. Wang determined rings of the forms
Zp
α1
1
×· · ·×Zp
αr
r
and Zn[x]
〈xm〉 that have genus at most one [24, Theorems 3.5
and 3.11]. Further H.J. Wang and N.O. Smith obtained all commutative
rings whose zero divisor graph has genus at most one [23, Theorem 3.6.2].
Note that the zero divisor graph Γ(R) is a subgraph of AG(R). In
[7], it has been shown that for any reduced ring R that is not an integral
domain, AG(R) = Γ(R) if and only if R has exactly two distinct minimal
prime ideals [7, Theorem 3.6]. Note that using the proof of this result,
“adm-n4” 22:47 page #15
T. Tamizh Chelvam, K. Selvakumar 193
x1
x2 x3
x4
x5x6x7
x8x9
x10
x11
Figure 1. Graph G.
one can establish that for any reduced ring, AG(R) is complete if and
only if, Γ(R) is complete if and only if, R ∼= Z2 × Z2.
By a graph G = (V, E), we mean an undirected simple graph with
vertex set V and edge set E. A graph in which each pair of distinct
vertices is joined by the edge is called a complete graph. We use Kn to
denote the complete graph with n vertices. An r-partite graph is one
whose vertex set can be partitioned into r subsets so that no edge has
both ends in any one subset. A complete r-partite graph is one in which
each vertex is joined to every vertex that is not in the same subset. The
complete bipartite graph (2-partite graph) with part sizes m and n is
denoted by Km,n. If G = K1,n where n > 1, then G is a star graph. Pn
denotes the path of length n for n > 1. A graph G is said to be unicyclic
if it contains a unique cycle. Given a connected graph G, we say that a
vertex v of G is a cut vertex if G − v is disconnected. For a subset S of
vertices of G, the induced subgraph of G is the subgraph with vertex set
S together with edges whose both ends are in S and is denoted by < S >.
A block is a maximal connected subgraph of G having no cut vertices. A
result of Battle, Harary, Kodama, and Youngs states that the genus of a
graph is the sum of the genera of its blocks[6]. For example, the graph G
in Figure 1 has two blocks, both isomorphic to K3,3, and so has genus 2
[25, C. Wickham].
Throughout this paper, we assume that R is a finite commutative ring
with identity, Z(R) its set of zero-divisors and Nil(R) its set of nilpotent
elements, R× its group of units, Fq denotes the field with q elements, and
R∗ = R − {0}. The following results are useful for further reference in
this paper.
“adm-n4” 22:47 page #16
194Genus of the annihilator graph of a commutative ring
Theorem 1. [23, Theorem 3.5.1] Let (R,m) be a finite local ring which
is not a field. Then Γ(R) is planar if and only if R is isomorphic to one
of the following 29 rings:
Z4,
Z2[x]
〈x2〉
, Z9,
Z3[x]
〈x2〉
, Z8,
Z2[x]
〈x3〉
,
Z4[x]
〈x3, x2 − 2〉
,
Z2[x, y]
〈x2, xy, y2〉
,
Z4[x]
〈2x, x2〉
,
F4[x]
〈x2〉
,
Z4[x]
〈x2 + x + 1〉
, Z25,
Z5[x]
〈x2〉
, Z16,
Z2[x]
〈x4〉
,
Z4[x]
〈x2 − 2, x4〉
,
Z4[x]
〈x3 − 2, x4〉
,
Z4[x]
〈x3 + x2 − 2, x4〉
,
Z2[x, y]
〈x3, xy, y2 − x2〉
,
Z4[x]
〈x3, x2 − 2x〉
,
Z8[x]
〈x2 − 4, 2x〉
,
Z4[x, y]
〈x3, x2 − 2, xy, y2 − 2, y3〉
,
Z4[x]
〈x2〉
,
Z4[x, y]
〈x2, y2, xy − 2〉
,
Z2[x, y]
〈x2, y2〉
, Z27,
Z3[x]
〈x3〉
,
Z9[x]
〈x2 − 3, x3〉
,
Z9[x]
〈x2 + 3, x3〉
.
One can have the following theorem from Theorem 3.7 [15].
Theorem 2. [15, Theorem 3.7] Let R = F1 × · · · × Fn be a finite ring,
where each Fi is a field and n > 2. Then Γ(R) is planar if and only if R
is isomorphic to one of the following rings:
Z2 × F, Z3 × F, Z2 × Z2 × Z2, Z2 × Z2 × Z3,
where F is a finite field.
Theorem 3. [23, Theorem 3.5.2] Let (R,m) be a finite local ring which
is not a field. Then g(Γ(R)) = 1 if and only if R is isomorphic to one of
the following 17 rings:
Z49,
Z7[x]
〈x2〉
,
Z2[x, y]
〈x3, xy, y2〉
,
Z4[x]
〈x3, 2x〉
,
Z4[x, y]
〈x3, x2 − 2, xy, y2〉
,
Z8[x]
〈x2, 2x〉
,
F8[x]
〈x2〉
,
Z4[x]
〈x3 + x + 1〉
,
Z4[x, y]
〈2x, 2y, x2, xy, y2〉
,
Z2[x, y, z]
〈x, y, z〉2
,
Z32,
Z2[x]
〈x5〉
,
Z4[x]
〈x3 − 2, x5〉
,
Z4[x]
〈x4 − 2, x5〉
,
Z8[x]
〈x2 − 2, x5〉
,
Z8[x]
〈x2 − 2x + 2, x5〉
,
Z8[x]
〈x2 + 2x − 2, x5〉
.
Theorem 4. [11, 12, Theorem 6.3] 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.
“adm-n4” 22:47 page #17
T. Tamizh Chelvam, K. Selvakumar 195
Theorem 5. [3, Lemma 2.12] Let R be a finite commutative ring. If
Γ(R) has exactly one vertex adjacent to every other vertex and no other
adjacent vertices, then either R ∼= Z2 × F , where F is a finite field with
|F | > 3, or R is local with maximal ideal m satisfying R
m
∼= Z2, m3 = 0
and |m2| 6 2. Thus |Γ(R)| is either pn or 2n − 1 for some prime p and
integer n > 1.
Theorem 6. [3, Theorem 2.10] Let R be a finite commutative ring. If
Γ(R) is complete, then either R ∼= Z2 × Z2 or R is local with char R = p
or p2, and |Γ(R)| = pn − 1, where p is prime and n > 1.
Theorem 7. [3, Theorem 2.13] Let R be a finite commutative ring with
|Γ(R)| > 4. Then Γ(R) is a star graph if and only if R ∼= Z2 × F, where
F is a finite field. In particular, if Γ(R) is a star graph, then Γ(R) = pn
for some prime p and integer n > 0. Conversely, each star graph of order
pn can be realized as Γ(R).
Theorem 8. [7, Theorem 3.6] Let R be a reduced commutative ring that
is not an integral domain. Then AG(R) = Γ(R) if and only if R has
exactly two distinct minimal prime ideals.
Theorem 9. [7, Theorem 3.10] Let R be a nonreduced commutative ring
with | Nil(R)∗| > 2 and let AGN (R) be the (induced) subgraph of AG(R)
with vertices Nil(R)∗. Then AGN (R) is complete.
2. Basic properties of annihilator graph
In this section, we state some basic observations of the annihilator
graph. Especially, we identify the annihilator ideal graph of small order
and in particular we list out all local rings R with |R| 6 7, for which the
annihilator graph AG(R) is complete.
Remark 1. Let R = F1 × F2 where F1 and F2 are finite fields. Then
R is reduced with exactly two distinct minimal prime ideals. Hence, by
Theorem 8, AG(R) ∼= Γ(R) = K|F ∗
1
|,|F ∗
2
|.
Remark 2. Let R 6= Z2 × Z2 be a local ring that is not a field and
|Z(R)∗| > 3. By Theorem 9, AG(R) is complete and hence gr(AG(R)) = 3.
On the other hand, let R be a finite commutative ring with identity but not
a field. Since R is finite, R ∼= R1 × · · · × Rn, where each Ri is a local ring.
If n > 3, then (1, 0, . . . , 0)− (0, 1, 0, . . . , 0)− (0, 0, 1, 0, . . . , 0)− (1, 0, . . . , 0)
is a cycle in AG(R) and hence gr(AG(R) = 3.
“adm-n4” 22:47 page #18
196Genus of the annihilator graph of a commutative ring
|Z(R)∗| Local ring R AG(R)
1 Z4, Z2[x]
〈x2〉
K1
2 Z9, Z3[x]
〈x2〉
K2
3 Z8, Z2[x]
〈x3〉
, Z4[x]
〈2x,x2−2〉
K3
3 F4[x]
〈x2〉
, Z4[x]
〈x2+x+1〉
K3
3 Z4[x]
<x,2>2 , Z2[x,y]
<x,y>2 K3
4 Z25, Z5[x]
〈x2〉
K4
Table 1.
Remark 3. Note that Γ(R) is a subgraph of AG(R). D.F. Anderson
et al., [2] gave all zero-divisor graphs of order 6 4. Using this, we give
in Table 1, all commutative local rings R for which |Z(R)∗| 6 4 and
AG(R) is complete. One can note that there are only two rings Z2 × F4
and Z3 × Z3 with |Z(R)∗| 6 4 which are not mentioned in Table 1 since
Γ(Z2 × F4) = K1,3 and Γ(Z3 × Z3) = K2,2 (refer Remark 1).
S. P. Redmond [13,14] has given all local commutative rings R with
|Z(R)∗| 6 7. Using the list given in [13, 14], Table 2 provides all finite
commutative local rings, for which 6 6 |Z(R)∗| 6 7 and AG(R) is com-
plete.
|Z(R)∗| Local Ring R AG(R)
6 Z49, Z7[x]
〈x2〉
K6
7 Z16, Z2[x]
〈x4〉
, Z4[x]
〈x2+2〉
, Z4[x]
〈x2+2x+2〉
K7
7 Z4[x]
〈x3−2,2x2,2x〉
, Z2[x,y]
〈x3,xy,y2〉
, Z8[x]
〈2x,x2〉
K7
7 Z4[x]
〈x3,2x2,2x〉
, Z4[x,y]
〈x2−2,xy,y2,2x,2y〉
, Z4[x]
〈x2+2x〉
K7
7 Z8[x]
〈2x,x2+4〉
, Z2[x,y]
〈x2,y2−xy〉
, Z4[x,y]
〈x2,y2−xy,xy−2,2x,2y〉
K7
7 Z4[x,y]
〈x2,y2,xy−2,2x,2y〉
, Z2[x,y]
〈x2,y2〉
, Z4[x]
〈x2〉
, K7
7 Z2[x,y,z]
〈x,y,z〉2 , Z4[x,y]
〈x2,y2,xy,2x,2y〉
, F8[x]
〈x2〉
, Z4[x]
〈x3+x+1〉
K7
Table 2.
“adm-n4” 22:47 page #19
T. Tamizh Chelvam, K. Selvakumar 197
Theorem 10. Let R be a finite commutative ring with identity that is
not a field. Then AG(R) is a tree if and only if R is isomorphic to one
of the following 5 rings:
Z4,
Z2[x]
〈x2〉
, Z9,
Z3[x]
〈x2〉
or Z2 × F,
where F is a finite field.
Proof. Since R is finite, R ∼= R1 × · · · × Rn, where each Ri is a local ring.
Suppose AG(R) is a tree. In view Remark 2, n 6 2.
Suppose R ∼= R1 × R2. If Z(R1)∗ 6= {0}, then there exist x1, y1 ∈
Z(R1)∗ such that x1y1 = 0 and |R×
1 | > 2. Let x = (0, 1), y = (x1, 0)
z = (y1, 1) and w = (1, 0). Then x, y, z, w ∈ Z(R)∗ and (x1, 1) ∈ ann(zw)
where as (x1, 1) is neither in ann(z) nor in ann(w). Hence x−y−z −w−x
is a cycle in AG(R), a contradiction. Hence R1 and R2 are fields and so
AG(R) ∼= K|R1|−1,|R2|−1. Since AG(R) is tree, |R1| = 2 or |R2| = 2 and
so R ∼= Z2 × F , where F is a finite field.
Suppose R ∼= R1. Since R is not a field, Z(R)∗ 6= {0}. Here R = R1
is a local ring and so AG(R) is complete. Hence by the assumption viz.,
AG(R) is a tree, we get that |Z(R)∗| 6 2. Hence R ∼= Z4, Z2[x]
〈x2〉
, Z9, or
Z3[x]
〈x2〉
.
The converse can be ascertained from Table 1 and Remark 1.
Theorem 11. Let R be a finite commutative ring with identity that is
not a field. Then AG(R) is unicyclic if and only if R is isomorphic to
one of the following 8 rings:
Z8,
Z2[x]
〈x3〉
,
Z4[x]
〈2x, x2 − 2〉
,
F4[x]
〈x2〉
,
Z4[x]
〈x2 + x + 1〉
,
Z4[x]
〈x, 2〉2
,
Z2[x, y]
〈x, y〉2
or Z3 × Z3.
Proof. Sufficient part follows from Table 1 and Remark 1.
Conversely, assume that AG(R) contains a unique cycle of length
ℓ > 3. Since R is finite, R ∼= R1 × · · · × Rn, where each Ri is a lo-
cal ring. Suppose n > 3. Let x1 = (1, 0, 0, . . . , 0), x2 = (0, 1, 0, . . . , 0),
x3 = (0, 0, 1, 0, . . . , 0), y1 = (0, 1, 1, 0 . . . , 0), y2 = (1, 0, 1, 0 . . . , 0). Then
x1, x2, x3, y1, y2 ∈ Z(R)∗ and x1−x2−x3−x1 as well as x1−y1−y2−x2−x1
are two distinct cycles in AG(R), a contradiction. Hence n 6 2.
Case 1. Suppose n = 2. If Z(R1) 6= {0}, then there exist x, y ∈ Z(R1)∗
such that xy = 0. Since R1 is local, R1 contains at least one unit u1 ∈ R×
1
“adm-n4” 22:47 page #20
198Genus of the annihilator graph of a commutative ring
apart from the identity. Then (x, 0) − (y, 1) − (1, 0) − (0, 1) − (x, 0) and
(x, 0)− (y, 1)− (u1, 0)− (0, 1)− (x, 0) are cycles in AG(R), a contradiction.
Hence R1 and R2 are fields and so AG(R) ∼= K|R1|−1,|R2|−1. Since AG(R)
is unicyclic, R1
∼= Z3 and R2
∼= Z3.
Case 2. Suppose n = 1. Here R is a local ring but not a field. By
Theorem 9, AG(R) is complete. Since AG(R) is unicyclic, |Z(R)∗| = 3
and by Table 1, R is isomorphic to one of the following rings: Z8, Z2[x]
〈x3〉
,
Z4[x]
〈2x,x2−2〉
, F4[x]
〈x2〉
, Z4[x]
〈x2+x+1〉
, Z4[x]
〈x,2〉2 , Z2[x,y]
〈x,y〉2 .
Note that any complete graph is a split graph with any single vertex
as an independent set and all the other vertices induce a clique. Hence, if
R is a local ring, then AG(R) is a split graph. Now, we characterize all
nonlocal rings R for which AG(R) is a split graph.
Theorem 12. Let R be a finite commutative nonlocal ring with identity
and |Z(R)∗| > 2. Then AG(R) is a split graph if and only if R ∼= Z2 × F ,
where F is a finite field.
Proof. Suppose R = Z2 × F , where F is a finite field. By Remark 1,
AG(R) = K1,|F ∗| and hence AG(R) is a split graph.
Conversely, suppose AG(R) is a split graph. Since R is finite, R ∼=
R1 × · · · × Rn where each Ri is local for 1 6 i 6 n and n > 2. If n > 3,
then (1, 0, 0, . . . , 0) − (0, 1, 0, . . . , 0) − (1, 0, 1, 0 . . . , 0) − (0, 1, 1, 0, . . . , 0) −
(1, 0, 0, . . . , 0) is a cycle of length 4 in AG(R) and by Theorem 4, AG(R)
is not split, a contradiction. Hence n = 2.
If Z(R1) 6= {0}, then there exists an element x ∈ Z(R1)∗ such that
xy = 0 for some y ∈ Z(R1)∗ and so (x, 0) − (0, 1) − (1, 0) − (y, 1) − (x, 0)
is a cycle of length 4 in AG(R), a contradiction. Hence R1, R2 are fields
and so AG(R) ∼= K|R1|−1,|R2|−1. Since AG(R) is split, either |R1| − 1 = 1
or |R2| − 1 = 1 and so R ∼= Z2 × F where F is a finite field.
Theorem 13. Let R be a finite commutative ring with identity that is
not a field. Then
(i) gr(AG(R)) = ∞ if and only if R ∼= Z4, Z2[x]
〈x2〉
, Z9, Z3[x]
〈x2〉
, or Z2 × F ,
where F is a finite field;
(ii) gr(AG(R)) = 4 if and only if R ∼= Z4 × Z2, Z2[x]
〈x2〉
× Z2, or F1 × F2,
where F1, F2 are finite fields with |F1| > 3 and |F2| > 3;
(iii) gr(AG(R)) = 3 if and only if R is not isomorphic to the rings in
(i) and (ii).
Proof. (i) Suppose gr(AG(R)) = ∞. Then AG(R) contains no cycles and
so AG(R) is a tree. Remaining part of the proof follows from Theorem 10.
“adm-n4” 22:47 page #21
T. Tamizh Chelvam, K. Selvakumar 199
(ii) Suppose gr(AG(R)) = 4. By Remark 2, R cannot be local. Also
|Z(R)∗| > 4. Let R ∼= R1 × · · · × Rn where each Ri is a local ring. If
n > 3, by Remark 2, gr(AG(R)) = 3, a contradiction and hence n = 2.
Suppose R is not reduced. Then |Z(Ri)
∗| 6= {0} for some i. If
|Z(R1)∗| > 2, then there exist x, y ∈ Z(R1)∗ such that x 6= y and
xy = 0. From this, we get that (x, 0) − (y, 0) − (0, 1) − (x, 0) is a cycle
of length 3 in AG(R), a contradiction. Thus |Z(Ri)
∗| 6 1 for i = 1, 2. If
|Z(Ri)
∗| = 1 for i = 1, 2, then Ri
∼= Z4 or Z2[x]
〈x2〉
. Suppose R ∼= Z4 × Z4,
then (2, 0)− (0, 1)− (2, 2) is a cycle in AG(R). Note that all the remaining
cases produce the same AG(R). Hence in all the cases gr(AG(R)) = 3, a
contradiction. This shows that |Z(R1)∗| = 1 or |Z(R2)∗| = 1.
If |Z(R1)∗| = 1 and |Z(R2)∗| = 0, then R1
∼= Z4 or Z2[x]
〈x2〉
and R2 is a
field. If |R2| > 3, then (2, 0) − (2, 1) − (2, x) − (2, 0) for some 1 6= x ∈ R∗
2
is a cycle of length three in AG(R), a contradiction. Hence |R2| = 2 and
so R2
∼= Z2 and so R ∼= Z4 × Z2 or Z2[x]
〈x2〉
× Z2.
Suppose R is reduced. Then R1 and R2 are fields and so AG(R) ∼=
K|R1|−1,|R2|−1. Since gr(AG(R)) = 4, |R1| > 3 and |R2| > 3.
Converse part of (ii) is trivial.
Part (iii) now follows directly from the above two cases.
Corollary 1. Let R be a finite commutative ring with identity that is
not a field. Then AG(R) is a complete bipartite graph if and only if R is
isomorphic to one of the following rings:
Z9,
Z3[x]
〈x2〉
, Z4 × Z2,
Z2[x]
〈x2〉
× Z2 or F1 × F2,
where F1 and F2 are finite fields.
Proof. We only need to prove the necessary part. Suppose AG(R) is a
complete bipartite graph. Then AG(R) does not contains a cycle of odd
length and so girth of AG(R) cannot be odd. By Theorem 13, gr(AG(R) =
4 or gr(AG(R)) = ∞. By the assumption that AG(R) is complete bipartite,
AG(R) ≇ K1. Hence R is isomorphic to one of the following rings: Z9,
Z3[x]
〈x2〉
, Z4 × Z2, Z2[x]
〈x2〉
× Z2, or F1 × F2, where F1 and F2 are fields.
3. Planar annihilator graphs
In this section, we characterize finite commutative rings R for which
AG(R) is planar. The following are known regarding the genus.
“adm-n4” 22:47 page #22
200Genus of the annihilator graph of a commutative ring
Lemma 1 ([26]). A connected graph G is planar if and only if G contains
no subdivision of either K5 or K3,3.
Lemma 2 ([26]). Let n be a positive integer and for real number x, ⌈x⌉
is the least integer that is the greater than or equal to x. Then g(Kn) =
⌈
(n−3)(n−4)
12
⌉
if n > 3. In particular, g(Kn) = 1 if n = 5, 6, 7.
Lemma 3 ([26]). Let m, n be positive integers and for real number x, ⌈x⌉
is the least integer that is the greater than or equal to x. Then g(Km,n) =
⌈
(m−2)(n−2)
4
⌉
if m, n > 2. In particular, g(K4,4) = g(K3,n) = 1 if n =
3, 4, 5, 6 and g(K5,4) = g(K6,4) = g(Km,4) = 2 if m = 7, 8, 9, 10.
Lemma 4 ([26, Euler formula]). If G is a finite connected graph with n
vertices, m edges, and genus g, then n − m + f = 2 − 2g, where f is the
number of faces created when G is minimally embedded on a surface of
genus g.
Theorem 14. Let (R,m) be a finite commutative local ring with identity.
Then AG(R) is planar if and only if R is isomorphic to one of the following
13 rings:
Z4,
Z2[x]
〈x2〉
, Z9,
Z3[x]
〈x2〉
, Z8,
Z2[x]
〈x3〉
,
Z4[x]
〈2x, x2 − 2〉
,
F4[x]
〈x2〉
,
Z4[x]
〈x2 + x + 1〉
,
Z4[x]
〈x, 2〉2
,
Z2[x, y]
〈x, y〉2
, Z25 or
Z5[x]
〈x2〉
.
Proof. By Lemma 1, AG(R) is planar if and only if AG(R) contains no
subdivision of either K5 or K3,3. Hence |Z(R)∗| 6 4. Now proof follows
from Table 1.
Theorem 15. Let R = R1 × · · · × Rn be a finite commutative nonlocal
ring, where each Ri is a local ring and n > 2. Then AG(R) is planar if
and only if R is isomorphic either of the following rings:
Z2 × F, Z3 × F, Z4 × Z2,
Z2[x]
〈x2〉
× Z2 or Z2 × Z2 × Z2,
where F is a finite field.
Proof. Note that AG(Z2 ×F ) = K1,|F ∗| and AG(Z3 ×F ) = K2,|F ∗| and so
by Lemma 3, they are planar. Since AG(Z4×Z2) = AG(Z2[x]
〈x2〉
× Z2) ∼= K2,3,
they are also planar. As per the embedding given in Figure 2,
AG(Z2 × Z2 × Z2) is planar.
“adm-n4” 22:47 page #23
T. Tamizh Chelvam, K. Selvakumar 201
(0, 1, 1)
(1, 0, 0)
(0, 0, 1) (0, 1, 0)
(1, 1, 0) (1, 0, 1)
Figure 2. AG(Z2 × Z2 × Z2).
Conversely assume that AG(R) is planar. Suppose n > 4. Let
x1 = (1, 0, 0, 0, . . . , 0), x2 = (0, 1, 0, 0, . . . , 0), x3 = (1, 1, 0, 0, . . . , 0),
y1 = (0, 0, 1, 0, . . . , 0), y2 = (0, 0, 0, 1, 0, . . . , 0), y3 = (0, 0, 1, 1, 0, . . . , 0).
Then Ω = {x1, x2, x3, y1, y2, y3} ⊆ Z(R)∗ and the subgraph of AG(R)
induced by Ω contains K3,3 as a subgraph, a contradiction. Hence n 6 3.
Case 1. n = 2. Suppose mi 6= {0} for all i = 1, 2. Then |Ri| > 4 and
|R×
i | > 2. Let ai, bi be two distinct elements in R∗
i other than identity for
i = 1, 2. Let d1 = (1, 0), d2 = (a1, 0), d3 = (b1, 0), g1 = (0, 1), g2 = (0, a2),
g3 = (0, b2) ∈ Z(R)∗. Then Ω1 = {d1, d2, d3, g1, g2, g3} ⊆ Z(R)∗ and
the subgraph of AG(R) induced by Ω1 contains K3,3 as a subgraph, a
contradiction. Hence mi = {0} for some i.
Without loss of generality, we assume that m2 = {0}. Then R2 is a
field.
Suppose m1 6= {0}. We claim that |m∗
1| = 1. If not, |m∗
1| > 2 and
|R×
1 | > 3. Note that |R2| > 2. For a, b ∈ m
∗
1 with ab = 0, and two distinct
units u1, u2 ∈ R×
1 , let z1 = (a, 1), z2 = (b, 1), z3 = (0, 1), w1 = (1, 0),
w2 = (u1, 0), w3 = (u2, 0) ∈ Z(R)∗. Then Ω2 = {z1, z2, z3, w1, w2, w3} ⊆
Z(R)∗ with z3wi = 0 in R for i = 1, 2, 3. Clearly z2 ∈ ann(z1w1), z2 /∈
ann(z1) ∪ ann(w1), z2 ∈ ann(z1w2), z2 /∈ ann(z1) ∪ ann(w2) and so z1 is
adjacent to both w1 and w2 in AG(R). Further z1 ∈ ann(z2w1), z1 /∈
ann(z2) ∪ ann(w1), z1 ∈ ann(z2w2), z1 /∈ ann(z2) ∪ ann(w2) and so z2 is
adjacent to both w1 and w2 in AG(R). From this, we observe that K3,3
is a subgraph of AG(R), a contradiction. Hence |m∗
1| = 1 and so R1
∼= Z4
or Z2[x]
〈x2〉
.
Suppose |R2| > 3. For d ∈ m
∗
1 with d2 = 0 and, 1 6= v1 ∈ R×
1 and
1 6= v2 ∈ R∗
2, let s1 = (d, 1), s2 = (d, v2), s3 = (0, 1), p1 = (d, 0), p2 =
“adm-n4” 22:47 page #24
202Genus of the annihilator graph of a commutative ring
(v1, 0), p3 = (1, 0). Then Ω3 = {s1, s2, s3, p1, p2, p3} ⊆ Z(R)∗ and, p1s1 =
p1s2 = 0 in R and s3pi = 0 in R for i = 1, 2, 3. Clearly s2 ∈ ann(s1p2),
s2 /∈ ann(s1) ∪ ann(p2), s2 ∈ ann(s1p3), s2 /∈ ann(s1) ∪ ann(p3) and
so s1 is adjacent to both p2 and p3 in AG(R). Also s1 ∈ ann(s2p2),
s1 /∈ ann(s2) ∪ ann(p2), s1 ∈ ann(s2p3), s1 /∈ ann(s2) ∪ ann(p3) and so s2
is adjacent to both p2 and p3 in AG(R). From this, we get that K3,3 is a
subgraph of AG(R), a contradiction. Hence |R2| = 2 and so R2
∼= Z2.
Suppose m1 = {0}. Then R1 is a field and by Remark 1, AG(R) ∼=
K|R1|−1,|R2|−1. By Lemma 3, R ∼= Z2 × F or Z3 × F , where F is a finite
field.
(1, 0, 0) (0, 1, 0) (1, 0, 1) (1, 1, 0)
(0, 0, 1) (0, 0, 2) (0, 1, 1) (1, 0, 2) (0, 1, 2)
Figure 3. AG(Z2 × Z2 × Z3).
Case 2. n = 3. Suppose |Ri| > 4 for some i. Without loss of generality, we
assume that |R1| > 4. Let u1, u2, u3 be three distinct nonzero elements
in R1. Let h1 = (u1, 0, 0), h2 = (u2, 0, 0), h3 = (u3, 0, 0), t1 = (0, 1, 0),
t2 = (0, 0, 1), t3 = (0, 1, 1) ∈ Z(R)∗. Then hitj = 0 for all i, j = 1, 2, 3
and so K3,3 is a subgraph of AG(R), a contradiction. Hence |Ri| 6 3 is
field for i = 1, 2, 3 and so Ri
∼= Z2 or Z3 for i = 1, 2, 3. Since Γ(R) is a
subgraph of AG(R) and AG(R) is planar, by Theorem 2, the possibilities
for R are Z2 × Z2 × Z2 or Z2 × Z2 × Z3. Note that the edges in dark
lines of Figure 3 constitute a subdivision of K3,3 and so by Lemma 1,
AG(Z2 × Z2 × Z3) is not planar. Thus R ∼= Z2 × Z2 × Z2.
4. Genus of AG(R)
In this section, we characterize isomorphism classes of finite commuta-
tive rings R whose AG(R) has genus one. First, let us characterize finite
commutative local rings R for which genus of AG(R) is one.
Theorem 16. Let (R,m) be a finite commutative local ring with identity
that is not a field. Then g(AG(R)) = 1 if and only if R is isomorphic to
“adm-n4” 22:47 page #25
T. Tamizh Chelvam, K. Selvakumar 203
one of the following 22 rings:
Z49,
Z7[x]
〈x2〉
, Z16,
Z2[x]
〈x4〉
,
Z4[x]
〈x4, x2 − 2〉
,
Z2[x]
〈x3 − 2, x4〉
,
Z4[x]
〈x4, x3 + x2 − 2〉
,
Z2[x]
〈x3, x2 − 2x〉
,
Z2[x, y]
〈x3, xy, y2 − x2〉
,
Z8[x]
〈x2 − 4, 2x〉
,
Z4[x, y]
〈x3, xy, x2 − 2, y2 − 2, y3〉
,
Z4[x]
〈x2〉
,
Z4[x, y]
〈x2, y2, xy − 2〉
,
Z2[x, y]
〈x2, y2〉
,
Z2[x, y]
〈x2, y2, xy〉
,
Z4[x]
〈x3, 2x〉
,
Z4[x, y]
〈x3, x2 − 2, xy, y2〉
,
Z8[x]
〈x2, 2x〉
,
F8[x]
〈x2〉
,
Z4[x]
〈x3 + x + 1〉
,
Z4[x, y]
〈2x, 2y, x2, y2, xy〉
,
Z2[x, y, z]
〈x, y, z〉2
.
Proof. By Theorem 9, AG(R) is complete. By Lemma 2, 5 6 |Z(R)∗| 6 7.
Note that there is no local ring R with |Z(R)∗| = 5. Now the proof follows
from Table 2.
Remark 4. Note that if R ∼= R1 × · · · × Rn is a commutative ring with
identity, where each (Ri,mi) is a local ring with mi 6= {0} and n > 2,
then K5,6 is a subgraph of AG(R) and hence g(AG(R)) > 2. Thus if
g(AG(R)) = 1, then one of the components Ri must be a field.
Theorem 17. Let R = R1 × · · · × Rn be a finite commutative nonlocal
ring, where each Ri is a local ring and n > 2. Then g(AG(R)) = 1 if and
only if R is isomorphic to one of the following 7 rings:
F4 × F4, F4 × Z5, Z5 × Z5, F4 × Z7, Z4 × Z3,
Z2[x]
〈x2〉
× Z3 or Z2 × Z2 × Z3.
Proof. Assume that g(AG(R)) = 1.
Suppose n > 4. Let
x1 = (1, 1, 0, 0, . . . , 0), x2 = (0, 1, 0, 1, . . . , 1), x3 = (0, 0, 1, 0, . . . , 0),
x4 = (0, 1, 1, 0, . . . , 0), x5 = (1, 0, 1, . . . , 1), x6 = (1, 1, 0, 1, . . . , 1),
x7 = (1, 1, 1, 0, . . . , 0), x8 = (1, 0, 1, 0, . . . , 0), x9 = (0, 0, 0, 1, . . . , 1),
x10 = (1, 0, 0, 1, . . . , 1), x11 = (0, 1, . . . , 1).
Then Ω = {x1, . . . , x11} ⊆ Z(R)∗ with x1x3 = x2x3 = x3x6 = x1x9 =
x7x9 = x8x9 = 0. Clearly x5 ∈ ann(x1x4), x5 /∈ ann(x1) ∪ ann(x4),
x11 ∈ ann(x1x5), x11 /∈ ann(x1)∪ann(x5), x5 ∈ ann(x2x4), x5 /∈ ann(x2)∪
“adm-n4” 22:47 page #26
204Genus of the annihilator graph of a commutative ring
ann(x4), x7 ∈ ann(x2x5), x7 /∈ ann(x2) ∪ ann(x5), x4 ∈ ann(x5x6),
x4 /∈ ann(x5) ∪ ann(x6), x5 ∈ ann(x6x4), x5 /∈ ann(x6) ∪ ann(x4), x11 ∈
ann(x1x10), x11 /∈ ann(x1) ∪ ann(x10), x5 ∈ ann(x1x11), x5 /∈ ann(x1) ∪
ann(x11), x11 ∈ ann(x8x10), x11 /∈ ann(x8) ∪ ann(x10), x6 ∈ ann(x11x8),
x6 /∈ ann(x11) ∪ ann(x8), x10 ∈ ann(x11x7), x10 /∈ ann(x11) ∪ ann(x7),
x11 ∈ ann(x10x7) and x11 /∈ ann(x10) ∪ ann(x7). From all the above ob-
servations, Note that the subgraph induced by Ω in AG(R) contains G
given in Figure 1 as a subgraph and so g(AG(R)) > 2. Hence n 6 3.
Case 1. n = 3. Suppose R1 and R2 are not fields. Then as mentioned
in Remark 4, K5,6 is a subgraph of AG(R) and so g(AG(R)) > 2, a
contradiction. Hence at least two of the components Ri, (1 6 i 6 3) must
be fields. Without loss of generality, let us assume that R2 and R3 are fields.
Then |R2| > 2 and |R3| > 2. Suppose R1 is not a field. Then |m1| > 2
and |R×
1 | > 2. For z ∈ m
∗
1 with ann(z) = m1 and u ∈ R×
1 and 1 6= u,
let x1 = (z, 1, 0), x2 = (0, 1, 0), x3 = (z, 0, 0), x4 = (1, 0, 0), x5 = (u, 0, 0),
x6 = (0, 0, 1, ), x7 = (0, 1, 1), x8 = (z, 1, 1), x9 = (1, 0, 1), x10 = (z, 0, 1),
x11 = (u, 0, 1) and Ω1 = {x1, . . . , x11}. Without much difficulty, one can
check that the subgraph induced by Ω1 in AG(R) contains G given in
Figure 1 and so g(AG(R)) > 2, a contradiction. Hence R1 must be a field.
Since AG(R) is non-planar, by Theorem 15, R ≇ Z2 × Z2 × Z2.
Suppose at least two of the components Ri(i 6 i 6 3) contain at
least three elements. Without loss of generality, assume that |R2| > 3
and |R3| > 3. For 1 6= a ∈ R∗
3 and 1 6= b ∈ R∗
2, let x1 = (1, 1, 0), x2 =
(0, 1, 0), x3 = (0, 0, 1), x4 = (0, 0, a), x5 = (1, 0, 1), x6 = (0, 1, 0), x7 =
(1, b, 0), x8 = (1, 0, 0), x9 = (0, 1, a), x10 = (0, 1, 1), x11 = (0, b, 1) and
Ω2 = {x1, . . . , x11}. Then the subgraph of AG(R) induced by Ω2 contains
G in Figure 1 and so g(AG(R)) > 2, a contradiction. Hence two of the
components Ri must be Z2. Without loss of generality, we assume that
R1 = R2 = Z2 and |R3| > 3.
Suppose |R3| > 4. Let u and v be two distinct non-zero elements in
R∗
3 other than identity. Let x1 = (1, 0, 0), x2 = (1, 0, u), x3 = (0, 1, 1),
x4 = (0, 1, v), x5 = (0, 1, u), x6 = (1, 0, v), x7 = (1, 1, 0), x8 = (0, 1, 0),
x9 = (0, 0, 1), x10 = (0, 0, u), x11 = (1, 0, v) and Ω3 = {x1, . . . , x11}. Then
the subgraph of AG(R) induced by Ω3 contains G given in Figure 1 and so
g(AG(R)) > 2, a contradiction. Hence R3 = Z3 and so R ∼= Z2 ×Z2 ×Z3.
Case 2. n = 2. If R1 and R2 are not fields, then as mentioned in Remark 4,
K5,6 is a subgraph of AG(R) and so g(AG(R)) > 2, a contradiction. Hence
one of the components Ri must be a field. Without loss generality, we
assume that R2 is a field and so |R2| > 2. By Theorem 15, R ≇ Z4 × Z2
and Z3[x]
〈x2〉
× Z2.
“adm-n4” 22:47 page #27
T. Tamizh Chelvam, K. Selvakumar 205
We claim that, if |m∗
1| > 2, then g(AG(R)) > 2.
Suppose |m∗
1| = 2. By the facts given in Table 1, R1
∼= Z9 or Z3[x]
〈x3〉
and hence |R×
1 | = 6. Let m
∗
1 = {a, b}, R×
1 = {u1, . . . , u6} and y1 =
(a, 1), y2 = (b, 1), y3 = (0, 1), x1 = (a, 0), x2 = (b, 0), x3 = (u1, 0),
x4 = (u2, 0), x5 = (u3, 0), x6 = (u4, 0), x7 = (u5, 0), x8 = (u6, 0). Then
Ω4 = {y1, y2, y3, x1, . . . , x8} ⊆ Z(R)∗ and the subgraph of AG(R) induced
by Ω4 contains K3,8. By Lemma 3, g(AG(R)) > 2, a contradiction.
Suppose |m∗
1| > 3. Then |R×
1 | > 4. Let d, e, f ∈ m
∗
1 such that de =
df = 0 and {v1, . . . , v4} ⊆ R×
1 . Consider Ω5 = {z1, . . . , z7, w1, . . . , w4}
⊆ Z(R)∗, where z1 = (d, 0), z2 = (e, 0), z3 = (f, 0), z4 = (v1, 0), z5 =
(v2, 0), z6 = (v3, 0), z7 = (v4, 0), w1 = (d, 1), w2 = (e, 1), w3 = (f, 1),
w4 = (0, 1). Then the subgraph induced by Ω5 of AG(R) contains K4,7
and by Lemma 3, g(AG(R)) > 2, a contradiction.
Hence we conclude that |m1| 6 2. From this R1
∼= Z4 or Z2[x]
〈x2〉
when
|m1| = 2 and R1 must be a field when |m1| = 1. By Theorem 15, R2 ≇ Z2
and so |R2| > 3.
Suppose |m1| = 2 and |R2| > 5. For a ∈ m
∗
1 with a2 = 0 and u1, u2 ∈
R×
1 , ej ∈ R∗
2, let x1 = (a, 0), x2 = (u1, 0), x3 = (u2, 0), x4 = (a, e1), x5 =
(a, e2), x6 = (a, e3), x7 = (a, e4), x8 = (0, e1), x9 = (0, e2), x10 = (0, e3),
x11 = (0, e4) and Ω6 = {x1, . . . , x11} ⊂ Z(R)∗. Then the subgraph
induced by Ω6 of AG(R) contains K3,8 and by Lemma 3, g(AG(R)) > 2,
a contradiction. Hence R2 is isomorphic to either Z3 or F4.
(2, 1)
(2, ω)
(2, ω2)
(1, 0) (2, 0) (3, 0)
(0, 1)
(0, ω)
(0, ω2)
Figure 4. AG(Z4 × F4) ∼= AG
(
Z2[x]
〈x
2〉 × F4
)
.
Consider the case that R2
∼= F4 = {0, 1, ω, ω2}. One can observe from
Figure 4 that K3,6 is a subgraph of AG(R). By Lemma 3, g(K3,6) = 1 and
hence one can fix an embedding of K3,6 on the surface of torus. By Euler’s
formula, there are 9 faces in the embedding of K3,6, say {S1, . . . , S9}. Let
ni be the length of the face Si. Note that
9
∑
i=1
ni = 36 and ni > 4 for
“adm-n4” 22:47 page #28
206Genus of the annihilator graph of a commutative ring
every i. Thus ni = 4 for every i. Let U = {(2, 1), (2, ω), (2, ω2)} ⊂ V (K3,6).
Further, the subgraph G of AG(R) induced by the vertices in U is K3,
E(G)∩E(K3,6) = ∅. Since K3 cannot be embedded in the torus along with
an embedding with only rectangles as faces, one cannot have an embedding
of G and K3,6 together in the torus. This implies that g(AG(R)) > 2.
Hence R2 ≇ F4 and so R is isomorphic to either Z4 × Z3 or Z2[x]
〈x2〉
× Z3.
Suppose |m1| = 1 and in this case both R1 and R2 are fields. Note
that AG(R) ∼= K|R1|−1,|R2|−1. Since g(AG(R)) = 1 and by Lemma 3, R
is isomorphic to one of the following rings F4 × F4, F4 × Z5, Z5 × Z5 or
F4 × Z7.
Converse follows from Remark 1, Lemma 3, Figure 5 and Figure 6.
(1, 0) (0, 2)
(3, 0) (2, 2)
(1, 0) (2, 0)
(2, 1)
c
c
d
d
e e
Figure 5. Embedding of AG(Z4 × Z3) ∼= AG
(
Z2[x]
〈x
2〉 × Z3
)
in S1.
(1, , 1, 0) (1, 0, 1)
(1, 0, 0) (0, 0, 1)
(0, 1, 1) (0, 1, 0)
(0, 0, 2)
(1, 0, 2)
(1, 0, 2)
(0, 1, 2)
(0, 1, 2)
a
a
c c
b b
Figure 6. Embedding of AG(Z2 × Z2 × Z3) in S1.
“adm-n4” 22:47 page #29
T. Tamizh Chelvam, K. Selvakumar 207
Acknowledgments
The authors thank the anonymous referees for their comments which
improved the presentation of the paper in many places.
References
[1] S. Akbari, H.R. Maimani and S. Yassemi, When a zero-divisor graph is planar or
a complete r-partite graph, J. Algebra 270 (2003), 169–180.
[2] D. F. Anderson, A. Frazier, A. Lauve, and P. S. Livingston, The zero-divisor graph
of a commutative ring II, Lecture Notes in Pure and Appl. Math. 202 (2001),
Marcel Dekker, New York, 61–72.
[3] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative
ring, J. Algebra 217 (1999), 434-447.
[4] T. Asir and T. Tamizh Chelvam, On the genus of generalized unit and unitary
Cayley graphs of a commutative ring, Acta Math. Hungar. 142 (2014), no. 2,
444–458, DOI: 10.1007/s10474-013-0365-1.
[5] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-
Wesley Publishing Company, 1969.
[6] J. Battle, F. Harary, Y. Kodama, J. W. T. Youngs, Additivity of the genus of a
graph, Bull. Amer. Math. Soc. 68 (1962), 565–568.
[7] A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra 42
(2014), no. 1, 108–121.
[8] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), 208–226.
[9] R. Belshoff, J. Chapman, Planar zero-divisor graphs, J. Algebra 316 (2007), no.
1, 471–480.
[10] G. Chartrand and L. Lesniak, Graphs and Digraphs, Wadsworth and Brooks/ Cole,
Monterey, CA, 1986.
[11] S. Földers and P.L. Hammer, Split Graphs, Proc. 8th Southeastern Conf. on
Combinatorics, Graph Theory and Computing (F. Koffman et al. eds.), Louisiana
State Univ., Baton Rouge, Louisianna, (1977), 311–315.
[12] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Second Edition,
Elsevier B.V., Amsterdam, 2004.
[13] S. P. Redmond, On zero-divisor graphs of small finite commutative rings, Discrete
Math. 307 (2007), 1155–1166.
[14] S. P. Redmond, Corrigendum to “On zero-divisor graphs of small finite commuta-
tive rings”, Discrete Math. 307 (2007), 2449–2452.
[15] N. O. Smith, Planar zero-divisor graphs, Int. J. Comm. Rings 2 (2002), no. 4,
177–188.
[16] T. Tamizh Chelvam and T. Asir, On the genus of the total graph of a commutative
ring, Comm. Algebra 41 (2013), 142–153.
[17] T. Tamizh Chelvam and S. Nithya, Crosscap of the ideal based zero-divisor graph,
Arab J. Math. Sci. 2 (2015), http://dx.doi.org/10.1016/j.ajmsc. 2015.01.003.
“adm-n4” 22:47 page #30
208Genus of the annihilator graph of a commutative ring
[18] T. Tamizh Chelvam and K. Selvakumar, Central sets in annihilating-ideal graph
of a commutative ring, J. Combin. Math. Combin. Comput. 88 (2014), 277–288.
[19] T. Tamizh Chelvam and K. Selvakumar, Domination in the directed zero-divisor
graph of ring of matrices, J. Combin. Math. Combin. Comput. 91 (2014), 155–163.
[20] T. Tamizh Chelvam and K. Selvakumar, On the intersection graph of gamma
sets in the zero-divisor, Dis. Math. Alg. Appl. 7 (2015), no. 1, # 1450067, doi:
10.1142/S1793830914500670.
[21] T. Tamizh Chelvam and K. Selvakumar, On the connectivity of the annihilating-
ideal graphs, Discussiones Mathematicae General Algebra and Applications 35
(2015), no. 2, 195 – 204, doi:10.7151/dmgaa.1241.
[22] T. Tamizh Chelvam, K. Selvakumar and V. Ramanathan, On the planarity of the k-
zero-divisor hypergraphs, AKCE International Journal of Graphs and Combinatorics
12 (2015), no. 2, 169–176.
[23] H. J. Wang and Neal O. Smith, Commutative rings with toroidal zero-divisor
graphs, Houston J. Math. 36 (2010), no. 1, 1–31.
[24] H. J. Wang, Zero-divisor graphs of genus one, J. Algebra 304 (2006), no. 2,
666–678.
[25] C. Wickham, Classification of rings with genus one zero-divisor graphs, Comm.
Algebra 36 (2008), 325–345.
[26] A. T. White, Graphs, Groups and Surfaces, North-Holland, Amsterdam, 1973.
Contact information
T. Tamizh Chelvam,
K. Selvakumar
Department of Mathematics
Manonmaniam Sundaranar University
Tirunelveli 627 012, Tamil Nadu, India
E-Mail(s): tamche59@gmail.com,
selva_158@yahoo.co.in
Received by the editors: 06.10.2015
and in final form 17.07.2016.
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