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AKCE International Journal of Graphs and Combinatorics (
)
–
www.elsevier.com/locate/akcej
Genus of total graphs from rings: A survey
T. Tamizh Chelvam a , ∗, T. Asir b
a Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli 627 012, Tamil Nadu, India
Received 2 January 2017; received in revised form 6 October 2017; accepted 6 October 2017
Available online xxxx
Abstract
Let R be a commutative ring. The total graph TΓ (R) of R is the undirected graph with vertex set R and two distinct vertices x
and y are adjacent if x + y is a zero divisor in R. In this paper, we present a survey of results on the genus of TΓ (R) and three of
its generalizations.
c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND
⃝
Keywords: Commutative ring; Total graph; Cayley graph; Genus; Planar
1. Introduction and preliminaries
The study of algebraic structures, using tools of graph theory, has become an exciting topic of research during the
last two decades. There are so many ways to associate a graph with an algebraic structure. Some of them to mention
are Cayley graphs from groups, non-commuting graph of a group, power graph of a finite group, zero-divisor graph
of a ring, total graph of a ring, unit graph of a ring, co-maximal graph of a ring, annihilating ideal graph of a ring
and torsion graph over modules. The study of these graphs constructed out of commutative rings deals with interplay
between the algebraic properties of a commutative ring and the graph theoretical properties of the corresponding
graph. Throughout this paper R denotes a commutative ring, Z (R) denotes the set of all zero-divisors in R and U (R)
denotes the set of all units in R.
The idea of associating a graph with zero-divisors of a commutative ring was introduced by Beck [1] in 1988.
In 1999, Anderson and Livingston modified the definition of Beck. There after, many research articles have been
published on the zero-divisor graph of commutative rings. In variation to the concept of zero- divisor graph, Anderson
and Badawi [2] introduced the total graph of a commutative ring. The total graph of a commutative ring R, denoted
by TΓ (R), is the undirected graph whose vertices are the elements in R and two distinct vertices x and y are adjacent
if x + y ∈ Z (R). In recent years, many research articles have been published on the total graph of commutative
rings [2–6].
Peer review under responsibility of Kalasalingam University.
∗ Corresponding author.
E-mail addresses: tamizhchelvam@msuniv.ac.in (T.T. Chelvam), asirjacob75@gmail.com (T. Asir).
https://doi.org/10.1016/j.akcej.2017.10.002
c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license
0972-8600/⃝
Please cite this article in press as: T.T. Chelvam, T. Asir, Genus of total graphs from rings: A survey, AKCE International Journal of Graphs and Combinatorics (2017),
https://doi.org/10.1016/j.akcej.2017.10.002.
2
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)
–
Akhtar et al. [7] defined the unitary Cayley graph of R, denoted by Cay(R, U (R)), as the graph whose vertex
set is R and two distinct vertices x and y are adjacent if x − y ∈ U (R). Khashyarmanesh et al. [8] introduced
the graph Γ (R, G, S) where G is a multiplicative subgroup of U (R) and S is a non-empty subset of G such that
S −1 = {s −1 : s ∈ S} ⊆ S. The graph Γ (R, G, S) is the simple graph with vertex set R and two distinct vertices x and
y are adjacent if there exists s ∈ S such that x + sy ∈ G.
Note that all these graphs are not isomorphic. But there are relations between these graphs. For instance, if S = {1},
then Γ (R, U (R), S) is same as the unit graph G(R) and if S = {−1}, then Γ (R, U (R), S) is the unitary Cayley graph
Cay(R, U (R)) of R. Further if R is finite, then the complement TΓ (R) of the total graph TΓ (R) is nothing but the unit
graph G(R). In case of an infinite ring, G(R) is a subgraph of TΓ (R).
For non-negative integers g and k, let Sg denote the sphere with g handles and Nk denote a sphere with k crosscaps
attached to it. It is well-known that every connected compact surface is homeomorphic to Sg or Nk for some nonnegative integers g and k. The genus of a graph G, denoted by g(G), is the minimum integer n such that the graph
can embedded in Sn . Similarly, crosscap number (nonorientable genus) g̃(G) is the minimum k such that G can be
embedded in Nk . A graph G is called planar if g(Γ (G)) = 0 and toroidal if g(Γ (G)) = 1.
Moreover, a nonorientable genus 1 graph is called a projective graph. Note that if H is a subgraph of a graph G,
then g(H ) ≤ g(G) and g̃(H ) ≤ g̃(G). An undirected graph is an outerplanar graph if it can be drawn in the plane
without crossings in such a way that all of the vertices belong to the unbounded face of the drawing. Clearly, every
outerplanar graph is planar. For details on the notion of embedding of graphs in a surface, we refer to White [9].
In this paper, we present results concerning the genus of the total and the generalized total graphs of commutative
rings. In Section 2, we present results on the characterization of all finite commutative rings R for which the genus
of the total graph of R is at most 2. More specifically, we present results on characterization of planar, toroidal and
projective total graphs and these are found in [3–5]. In Section 3, we present results on the genus of the complement
of the total graph of commutative rings and generalized total graphs [8,10–14].
2. Genus of total graph
In this section, first we discuss some bounds on the genus of the total graph of a commutative ring. Afterwards, we
present the characterization of all commutative rings R according to value of the genus of the total graph TΓ (R). The
result stated below gives a lower bound for the genus of the total graph of a class of rings.
Lemma 2.1 ([4, Lemma 1.3]). Let Fq denote the field with q elements. Then the total graph
⌈ of F2 × ⌉Fq is isomorphic
.
to K 2 × K q . Furthermore, for any positive integer m and q > 2, g(TΓ (F2m × Fq )) ≥ 2m (q−3)(q−4)
12
Theorem 2.2 ([4, Theorem 1.4]). For any positive integer m, there are finitely many finite commutative rings R whose
total graph has genus m.
The next theorem gives a lower bound for the genus of the total graph of a finite commutative ring.
Theorem 2.3 ([5, Theorem 3.2]). Let R be a finite commutative ring with identity, I be an ideal contained in Z (R),
|I | = λ ≥ 3 and | RI | = µ. Then the following are true:
⌉
⎧ ⌈
(λ − 3)(λ − 4)
⎪
⎪
if 2 ∈ I ;
⎨µ
12
⌈
⌉
⌈
⌉
g(TΓ (R)) ≥
⎪
(λ − 3)(λ − 4)
µ − 1 (λ − 2)2
⎪
⎩
+(
)
if 2 ̸∈ I .
12
2
4
Moreover, if R is local, then the equality holds good.
R
Note that if we take I as an ideal with | RI | = min{| m
| : m is a maximal ideal of R}, then the bound in Theorem 2.3 is
better. The following example shows that in various cases, the bounds for the genus of the total graph of a commutative
ring given in Theorem 2.3 are sharp.
⌈
⌉
Example 2.4. (a) Let R = Z6 and I = {0, 2, 4}. By Theorem 2.10, g(TΓ (R)) = 0 and so g(TΓ (R)) = µ (λ−3)(λ−4)
.
12
⌈
⌉
(λ−3)(λ−4)
(b) Let R = Z10 and I = {0, 2, 4, 6, 8}. By Theorem 2.12, g(TΓ (R)) = 2 and so g(TΓ (R)) = µ
.
12
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)
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⌈ (c) Let R
⌉ = Z3 ×⌈Z3 and
⌉I = {(0, 0), (0, 1), (0, 2)}. By Theorem 2.11, g(TΓ (R)) = 1 and therefore g(TΓ (R)) =
µ−1
(λ−3)(λ−4)
(λ−2)2
+( 2 )
.
12
4
Now we present some better lower bounds for the genus of the total graph of some special classes of commutative
rings.
Theorem 2.5 ([5, Theorem 3.7]). Let R be a finite commutative ring with identity, I be an ideal with maximal
cardinality among the proper ideals in R such that I = λ, | RI | = µ and 2 ≤ λ ̸≡ 5, 9 (mod 12). If Z (R) is not an
ideal of R and 2 ∈ I , then
⌉
⎧ ⌈
µ (λ − 2)(λ − 3)
⎪
⎪
if µ is even;
⎨
2
6
)⌈
⌉ ⌈
⌉
g(TΓ (R)) ≥ (
µ−1
(λ − 2)(λ − 3)
(λ − 3)(λ − 4)
⎪
⎪
⎩
+
if µ is odd.
2
6
12
Theorem 2.6 ([5, Theorem 3.2]). Let R be a finite commutative ring with identity, I be an ideal contained in Z (R),
|I | = λ ≥ 3 and | RI | = µ. Then the following are true:
⌉
⎧ ⌈
(λ − 3)(λ − 4)
⎪
⎪µ
if 2 ∈ I ;
⎨
12
⌈
⌈
⌉
⌉
g(TΓ (R)) ≥
⎪
(λ − 3)(λ − 4)
µ − 1 (λ − 2)2
⎪
⎩
)
+(
if 2 ̸∈ I .
12
2
4
Moreover, if R is local, then the equality holds true.
Recall that every finite commutative ring is a direct product of local rings. Using this, another lower bound for the
genus of the total graph of a finite commutative ring was obtained by Tamizh Chelvam and Asir [5]. Moreover this
lower bound on the genus of a direct product depends on the order in which one decomposes the total graph. Further,
using a relationship between the total graph of a direct product of rings and the Cartesian products total graphs of the
components, the following was proved.
Theorem 2.7 ([5, Theorem 3.8]). Let R be a finite commutative ring and R ∼
= R1 × R2 × · · · × Rm where
each Ri is a finite local ring. Let G i = TΓ (Ri ), ki = g(G i ), ℓ1 = max{|R1 |k2 + k1 , |R2 |k1 + k2 } and ℓ j =
∑j
max{( i=1 |Ri |)k j+1 + ℓ j−1 , |R j |ℓ j−1 + k j+1 } for j = 2, . . . , m − 1. Then ℓm−1 ≤ g(TΓ (R)).
The following example illustrates the use of Theorem 2.7.
Z3 [x]
Example 2.8. (a) Consider the ring R = Z9 × Z(x3 [x]
2 ) . By Theorem 2.11, k 1 = g(TΓ (Z9 )) = 1 and k 2 = g(TΓ ( (x 2 ) )) =
1. Thus by Theorem 2.7, g(TΓ (R)) ≥ 10.
(b) Let R = Z9 × Z9 × Z9 . Then ki = g(TΓ (Z9 )) = 1 for i = 1, 2, 3. Therefore by Theorem 2.7, ℓ1 = 10 and
ℓ2 = max{28, 91} = 91 and so g(TΓ (R)) ≥ 91.
The following theorem gives an upper bound for the genus of the total graph of a commutative ring.
⋃m
Theorem 2.9 ([5, Theorem 3.10]). Let R be a finite commutative ring and Z (R) = i=1
Pi where Pi ’s are ideals
Ri
of R. Let |Pi | = αi and | Pi | = βi for i = 1, . . . , m. Suppose that 2 ∈ Pt for all 1 ≤ t ≤ j and 2 ̸∈ Pt for all
j + 1 ≤ t ≤ m, where j = 0 if 2 ̸∈ Z (R). Then
⌈
⌉
j
∑
(αt − 3)(αt − 4)
g(TΓ (R)) ≤ (m − 1)(|R| − 1) +
βt
+
12
t=1
)⌈
⌉ ⌈
⌉}
m {(
∑
βt − 1
(αt − 2)2
(αt − 3)(αt − 4)
+
.
2
4
12
t= j+1
⌈
⌉
i −4)
(If αi = 2 for some i then we take (αi −3)(α
= 0.)
12
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https://doi.org/10.1016/j.akcej.2017.10.002.
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Now, we state results on the characterization of all commutative rings whose total graph has genus less than or
equal to 2. Maimani et al. [4] determined all finite commutative rings R for which TΓ (R) is a planar or toroidal graph.
The relevant results in this regard are stated below.
Theorem 2.10 ([4, Theorem 1.5]). Let R be a finite commutative ring such that TΓ (R) is planar. Then the following
hold:
(i) If R is local ring, then R is a field or R is isomorphic to the one of the 9 following rings:
Z2 [x] Z2 [x] Z2 [x, y] Z4 [x]
Z4 [x]
F4 [x]
Z4 [x]
,
Z4 ,
,
,
,
, Z8 , 2 , 2
;
(x 2 ) (x 3 ) (x, y)2 (2x, x 2 ) (2x, x 2 − 2)
(x ) (x + x + 1)
(ii) If R is not local ring, R is isomorphic to Z2 × Z2 or Z6 .
Next result states the characterization of all commutative rings whose total graph has genus 1.
Theorem 2.11 ([4, Theorem 1.6]). Let R be a finite commutative ring such that TΓ (R) is toroidal. Then the following
statements hold:
(i) If R is local ring, then R is isomorphic to Z9 , or Z(x3 [x]
2) ;
(ii) If R is not local ring, then R is isomorphic to one of the following rings:
Z2 [x]
, Z2 × Z2 × Z2 .
Z2 × F4 , Z3 × Z3 , Z2 × Z4 , Z2 ×
(x 2 )
In the following theorem, a characterization of all commutative rings whose total graph has genus two is obtained
by Tamizh Chelvam and Asir [5].
Theorem 2.12 ([5, Theorem 4.3]). Let R be a finite commutative ring. Then g(TΓ (R)) = 2 if and only if R is
isomorphic to either Z10 or Z3 × F4 .
The projective total graph of commutative rings was studied by Khashyarmanesh [15]. Recall that a nonorientable
genus 1 graph is called a projective graph. The following theorem provides a characterization of the local ring R
whose TΓ (R) is projective.
Theorem 2.13 ([15, Theorem 2.3]). Let R be a finite local ring. Then the total graph TΓ (R) is projective if and only
if R is isomorphic to one of the rings Z9 or Z(x3 [x]
2) .
Next theorem provides a characterization of all non-local rings R whose TΓ (R) is projective.
Theorem 2.14 ([15, Theorem 2.4]). Let R be a finite non-local ring. Then the total graph TΓ (R) is projective if and
only if R is isomorphic to Z3 × Z3 .
Theorems 2.13 and 2.14, in conjunction with Theorem 2.11 show that if R is a ring whose total graph is projective,
then TΓ (R) is also toroidal. This leads the following conjecture.
Conjecture. The inequality g̃(TΓ (R)) ≥ g(TΓ (R)) always holds for any commutative ring R.
3. Genus of generalized total graphs
In this section, we discuss about the genus of generalized total graphs of commutative rings. There are three major
types of generalizations available and first let us present the definitions of these three generalizations.
Definition 3.1.
(1) (Abbasi and Habibi [16]) Let R be a commutative ring and I be its proper ideal. Let S(I ) be the set of all
elements of R that are not prime to I ; i.e., S(I ) = {a ∈ R : ra ∈ I for some r ∈ R \ I }. The total graph of
R with respect to I , denoted by T (Γ I (R)), is the undirected graph with all elements of R as vertices, and for
distinct x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ S(I ). In the case I = {0}, the graph
T (Γ I (R)) is the total graph of R.
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(2) (Barati et al. [17]) Let R be a commutative ring and S be a multiplicatively closed subset of R. Define a simple
graph, denoted by Γ S (R), with all elements of R as vertices, and two distinct vertices x, y ∈ R are adjacent if
and only if x + y ∈ S. If we take S = Z (R), then Γ S (R) = TΓ (R).
(3) (Anderson and Badawi [18]) Let R be a commutative ring and H be a nonempty proper subset of R such that
R \ H is a saturated multiplicatively closed subset of R. The generalized total graph of R is the simple graph
GTH (R) with all elements of R as the vertices, and two distinct vertices x and y are adjacent if and only if
x + y ∈ H . When H = Z (R), we have that GTH (R) is the total graph of R.
Recently, Asir and Mano [19] obtained several bounds on the genus for the graph T (Γ I (R)). As a consequence
some of the results given in the previous section are generalized. We now provide a lower bound for the genus of
T (Γ I (R)) through the cardinality of a prime ideal P of R contained in S(I ). The following theorem can be considered
as an improved form of Theorem 2.3.
Theorem 3.2. Let R be a ring with the proper ideal I. Let P be a prime ideal of R contained in S(I ) such that
|P| = λ ≥ 3 and |R/P| = µ (λ and µ may be infinite cardinals also). Then
⌉
⎧ ⌈
(λ − 3)(λ − 4)
⎪
⎪µ
⎨
12
⌈
⌉ (
)⌈
⌉
γ (T (Γ I (R))) ≥
⎪
(λ − 3)(λ − 4)
µ−1
(λ − 2)2
⎪
⎩
+
12
2
4
if 2 ∈ P,
if 2 ̸∈ P.
Moreover, if S(I ) is an ideal of R, then the equality holds true.
Note that Theorem 3.2 is an effective tool to determine the genus of the total graph of a ring with respect to an
ideal.
Remark 3.3. Let R be a finite local ring. Then R has a unique prime ideal, which is nothing but the zero-divisors
Z (R). Therefore, S(I ) = Z (R) for every ideal I in R. Thus the second part of Theorem 3.2 gives us the exact value
of γ (T (Γ I (R))), whenever R is local. Thus, one can easily calculate the genus of the ideal based total graph of any
local ring. For instant, if R is either the local ring Z25 or Z(x5 [x]
2 ) , then λ = µ = 5 and so γ (T (Γ I (R))) = 7, and if R is
either the local ring Z49 or
Z7 [x]
,
(x 2 )
then λ = µ = 7 and so γ (T (Γ I (R))) = 22.
The next result exhibits a lower bound for the genus of T (Γ I (R)), where R is isomorphic to the direct product of
local rings, in terms of the cardinalities of the local rings. Moreover, any ideal of R is of the form direct product of (not
necessarily proper) ideals of local rings. It is worthwhile to mention that Artinian rings have such a decomposition to
local rings.
Theorem 3.4. For i = 1, 2, . . . , n (n ≥ 2), let Ri be finite local rings and Ii be ideals of Ri . Let R ∼
= R1 × · · · × Rn
ˆj | · · · |Rk |,
and I ∼
set
r
=
|R
|
· · · |R
= I1 ×· · ·× Ik × Rk+1 ×· · ·× Rn (1 ≤ k ≤ n). Let min{|R1 |, . . ., |Rk⌈|} = |R j | and
1
⌉
(r
−3)(r
−4)
ˆj | means that |R j | is removed. Then γ (T (Γ I (R))) ≥
if r ≥ 3 and γ (T (Γ I (R))) ≥ 0
where the notation |R
12
if r = 2.
Now, we state another lower bound for the ideal based total graph of a finite ring. One can see that Theorem 3.5 is
a slight generalization of Theorem 2.7.
Theorem 3.5. Let R be a finite ring and write R ∼
= R1 × · · · × Rn and I ∼
= I1 × · · · × In , where every Ri
is local and Ii a proper ideal of R. Set ki = γ (T (Γ Ii (Ri ))), ℓi = |Ri |, m 1 = max{ℓ1 k2 + k1 , ℓ2 k1 + k2 } and
m j = max{ℓ1 · · · ℓ j k j+1 + m j−1 , ℓ j+1 m j−1 + k j+1 } for j = 2, . . . , n − 1. Then γ (T (Γ I (R))) ≥ m n−1 .
Following theorem gives an upper bound for γ (T (Γ I (R))). Next we state results about the graph T (Γ I (R)) which
are natural generalizations of the corresponding results for the total graph in Theorem 2.9.
⋃m
Theorem 3.6. Let R be a finite ring and I be its proper ideal. Let S(I ) = i=1
Pi , where Pi ’s are prime ideals of R.
Let |Pi | = λi ≥ 3 and | PRi | = µi for i = 1, . . . , m. Rearrange Pi ’s such that 2 ∈ Pi for 1 ≤ i ≤ j and 2 ̸∈ Pi for
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j + 1 ≤ i ≤ m. Then
j
∑
{ ⌈
⌉}
(λi − 3)(λi − 4)
µ
γ (T (Γ I (R))) ≤ (m − 1)(|R| − 1) +
12
i=1,|λi |>2
{⌈
⌉ (
)⌈
⌉}
m
∑
(λi − 3)(λi − 4)
µi − 1
(λi − 2)2
+
+
.
12
2
4
i= j+1,|λ |>2
i
Finally, we provide an isomorphism relation between two total graphs T (Γ I (R)) and T (Γ J (S)) of two finite rings
R and S even when R ̸∼
= S. This result is an effective tool to compare the genus of ideal based total graphs of
same order rings. It is notable that if R and S are two rings with ideals I and J such that R ∼
= S and I ∼
= J,
Z [x]
∼
then clearly T (Γ I (R)) = T (Γ J (S)). However, the converse is not true in general. For example, Z4 ̸∼
= (x2 2 ) , whereas
Z [x]
T (Γ I (Z4 )) ∼
= T (Γ J ( 2 2 )) = K 2 ∪ K 2 for every ideals I and J .
(x )
Theorem 3.7. Let R ∼
= R1 × · · · × Rn and S ∼
= S1 × · · · × Sn be two finite rings, where Ri ’s and Si ’s are local. Let
∼
I = I1 × · · · × Ik × Rk+1 × · · · × Rn and J ∼
= J1 × · · · × Jk × Sk+1 × · · · × Sn be ideals of R and S respectively.
Then the following statements hold true:
(1) If for every 1 ≤ i ≤ k and k +1 ≤ j ≤ n, T (Γ Ii (Ri )) ∼
= T (Γ Ji (Si )) and |R j | = |S j |, then T (Γ I (R)) ∼
= T (Γ J (S)).
(2) If for every 1 ≤ i ≤ n, |Ri | = |Si | and |K Ri | = |K Si |, then T (Γ I (R)) ∼
= T (Γ J (S)).
4. Genus of the complement of total graph and its generalization
The complement TΓ (R) of the total graph TΓ (R) is a simple undirected graph with R as its vertex set, and two
distinct vertices x, y in TΓ (R) are adjacent if x + y ∈ Reg(R) [6]. The unit graph [20] of a ring R, denoted by G(R),
is the graph obtained by setting all the elements of R to be the vertices and defining distinct vertices x and y to be
adjacent if and only if x + y ∈ U (R). Note that if the ring is finite, then TΓ (R) is nothing but the unit graph. In the
case of an infinite ring, G(R) is a subgraph of TΓ (R), not necessarily a proper subgraph. For example, consider the
Q(x)
ring R = Q[x](+) Q[x]
. Then R is infinite with U (R) = Reg(R) and so TΓ (R) = G(R).
Note that Akhtar et al. [7] determined all finite commutative rings whose unitary Cayley graph has genus zero.
Also Ashrafi et al. [20] determined all finite commutative rings whose unit graph has genus zero. Further Wang [21]
obtained all commutative rings whose co-maximal graph has genus at most one. Khashyarmanesh et al. [8] determined
all commutative Artinian rings R for which Γ (R, U (R), S) is planar. The results in this regard are stated below.
Theorem 4.1 ([8, Theorem 3.7]). Let R be a commutative Artinian ring. Then Γ (R, U (R), S) is planar if and only if
one of the following conditions holds.
(i)
(ii)
(iii)
(iv)
(v)
R
R
R
R
R
∼
= Zℓ2 × T where ℓ ≥ 0 and T is isomorphic to one of the following rings: Z2 , Z3 , Z4 or
∼
= F4 ;
∼
= Zℓ2 × F4 where ℓ > 0 with S = {1};
∼
= Z5 with S = {1};
∼
= Z3 × Z3 with S = {(1, 1)}, S = {(1, −1)}, or S = {(−1, 1)}.
Z2 [x]
;
(x 2 )
The following theorem characterizes all commutative Artinian rings whose Γ (R, U (R), S) is toroidal.
Theorem 4.2 ([10, Theorem 4.2]). Let R be a commutative Artinian ring. Then Γ (R, U (R), S) is toroidal if and only
if one of the following conditions hold.
(i)
(ii)
(iii)
(iv)
(v)
R
R
R
R
R
∼
= Z5 with S ̸= {1};
∼
= Z7 ;
4 [x]
∼ Z8 , Z2 [x]
, Z2 [x,y] , Z4 [x] , or (x 2Z−2,2x)
;
=
(x 3 ) (x 2 ,x y,y 2 ) (x 2 ,2x)
∼
= Z2 × F4 with S ̸= {(1, 1)};
∼
= Z9 with S = {−1}, S = {2, 5}, or S = {−1, 2, 5};
Please cite this article in press as: T.T. Chelvam, T. Asir, Genus of total graphs from rings: A survey, AKCE International Journal of Graphs and Combinatorics (2017),
https://doi.org/10.1016/j.akcej.2017.10.002.
T.T. Chelvam, T. Asir / AKCE International Journal of Graphs and Combinatorics (
)
–
7
Z [x]
(vi) R ∼
= (x3 2 ) with S = {−1 + (x 2 )}, S = {−1 + x + (x 2 ), −1 − x + (x 2 )}, or S = {−1 + (x 2 ), −1 + x + (x 2 ), −1 −
x + (x 2 )};
∼ Z3 × Z3 with S = {(−1, −1)}, S = {(1, 1), (1, −1)}, S = {(1, 1), (−1, 1)}, S = {(−1, −1), (1, −1)}, or
(vii) R =
S = {(−1, −1), (−1, 1)};
(viii) R ∼
= Z10 with S = {1} or S = {−1};
(ix) R ∼
= Z12 with S = {1}, S = {−1}, S = {5}, S = {7}, S = {1, 7}, or S = {5, −1};
Z [x]
(x) R ∼
= Z3 × (x2 2 ) with S = {(1, 1 + (x 2 ))}, S = {(−1, 1 + x + (x 2 ))}, S = {(1, 1 + x + (x 2 ))}, S = {(−1, 1 + (x 2 ))},
S = {(1, 1 + (x 2 )), (1, 1 + x + (x 2 ))}, or S = {(−1, 1 + (x 2 )), (−1, 1 + x + (x 2 ))}.
Note that Γ (R, U (R), {1}) is nothing but the unit graph G(R). Using this fact and Theorem 4.2, now we enumerate
all commutative Artinian rings whose G(R) is toroidal and the same is given below.
Corollary 4.3. Let R be a commutative Artinian ring. Then the unit graph G(R) is toroidal if and only if R is
Z2 [x,y]
Z4 [x]
Z4 [x]
Z2 [x]
isomorphic to one of the rings: Z7 , Z8 , Z(x2 [x]
3 ) , (x 2 ,x y,y 2 ) , (x 2 ,2x) , (x 2 −2,2x) , Z10 , Z12 , and Z3 × (x 2 ) .
Note that the complement of the total graph TΓ (R) is same as the unit graph G(R) if R is finite and G(R)
is a subgraph of TΓ (R) if R is infinite. Using this fact and Theorem 4.2, we have the following corollary, which
characterizes all commutative rings for which TΓ (R) is toroidal.
Corollary 4.4. Let R be a commutative Artinian ring. Then TΓ (R) is toroidal if and only if R is isomorphic to one of
Z2 [x,y]
Z4 [x]
Z4 [x]
Z2 [x]
the rings: Z7 , Z8 , Z(x2 [x]
3 ) , (x 2 ,x y,y 2 ) , (x 2 ,2x) , (x 2 −2,2x) , Z10 , Z12 , and Z3 × (x 2 ) .
In view of the fact that Γ (R, U (R), {−1}) = Cay(R, U (R)), the corollary given below characterizes all
commutative rings whose unitary Cayley graph is toroidal.
Corollary 4.5. Let R be a commutative Artinian ring. The unitary Cayley graph Cay(R, U (R)) is toroidal if and only
Z3 [x]
Z2 [x,y]
Z4 [x]
Z4 [x]
if R is isomorphic to one of the rings: Z5 , Z7 , Z8 , Z(x2 [x]
3 ) , (x 2 ,x y,y 2 ) , (x 2 ,2x) , (x 2 −2,2x) , Z9 , (x 2 ) , Z2 × F4 , Z3 ×
Z3 , Z10 , Z12 , and Z3 ×
Z2 [x]
.
(x 2 )
The next result characterizes all commutative Artinian rings R whose Γ (R, U (R), S) has genus two.
Theorem 4.6 ([11, Theorem 3.5]). Let R be a commutative Artinian ring. Then g(Γ (R, U (R), S)) = 2 if and only if
one of the following holds:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
(xiii)
R
R
R
R
R
R
R
R
∼
= F8 ;
∼
= Z9 with S = {1}, S = {4, 7} or S = {1, 4, 7};
Z [x]
∼
= (x3 2 ) with S = {1 + (x 2 )}, S = {1 + x + (x 2 ), 1 − x + (x 2 )} or S = {1 + (x 2 ), 1 + x + (x 2 ), 1 − x + (x 2 )};
∼
= Z3 × Z3 with S = {(1, 1), (−1, −1)}, S = {(1, −1), (−1, 1)} or |S| ≥ 3;
∼
= Z10 with S ̸= {1} and S ̸= {−1};
∼
= Z3 × F4 with S = {(1, 1)} or S = {(−1, −1)};
∼ Z2 × Z2 × F4 with S ̸= {(1, 1, 1)};
=
Z [x]
Z [x,y]
Z [x]
∼
or Z2 × 2Z4 [x] ;
= Z2 × Z8 , Z2 × 2 3 , Z2 × 22 2 , Z2 × 24
(x )
(x ,x y,y )
(x ,2x)
(x −2,2x)
Z [x]
Z [x]
Z [x]
R∼
= Z4 × Z4 , (x2 2 ) × (x2 2 ) or Z4 × (x2 2 ) ;
∼ Z2 × Z3 × Z3 with S = (1, 1, 1), S = (1, 1, −1), S = (1, −1, 1) or S = (1, −1, −1);
R=
R∼
= Z2 × Z10 with S = {(1, 1)} or S = {(1, −1)};
R ∼
= Z2 × Z12 with S = {(1, 1)}, S = {(1, −1)}, S = {(1, 5)}, S = {(1, 7)}, S = {(1, 1), (1, 7)} or
S = {(1, 5), (1, −1)};
Z [x]
R ∼
= Z2 × Z3 × (x2 2 ) with S = {(1, 1, 1 + (x 2 ))}, S = {(1, −1, 1 + x + (x 2 ))}, S = {(1, 1, 1 + x + (x 2 ))},
S = {(1, −1, 1+(x 2 ))}, S = {(1, 1, 1+(x 2 )), (1, 1, 1+x +(x 2 ))} or S = {(1, −1, 1+(x 2 )), (1, −1, 1+x +(x 2 ))}.
Please cite this article in press as: T.T. Chelvam, T. Asir, Genus of total graphs from rings: A survey, AKCE International Journal of Graphs and Combinatorics (2017),
https://doi.org/10.1016/j.akcej.2017.10.002.
8
T.T. Chelvam, T. Asir / AKCE International Journal of Graphs and Combinatorics (
)
–
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Please cite this article in press as: T.T. Chelvam, T. Asir, Genus of total graphs from rings: A survey, AKCE International Journal of Graphs and Combinatorics (2017),
https://doi.org/10.1016/j.akcej.2017.10.002.
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