1. Graphs associated to commutative rings
Ayman Badawi
Department of Math.
American Univ. of Sharjah

2. References
Just type zero-divisor graph in the google bar, you will get at least published articles in well-known Journals. So there is no way I can state all of them!!!
The idea of zero-divisor graph of a commutative ring was initiated in this paper: I. Beck, Coloring of commutative rings, J. Algebra 116(1988), In this paper, the author was interested in coloring the vertices of such graph (he let the vertices of such graph to be all elements of R).
The zero-divisor graph as we know it now and studied by many authors was defined in this paper: David . F. Anderson and Paul. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217(1999),
Google Scholar (403 citation), Math Science Net (148 citation)

3. Zero-divisor graph in the sense of Anderson-Livingston paper*
Example: R= Z_6
Z(R)* = {2, 3, 4} 2
Star Graph, K_{1, 2}, girth = infinity, diameter = 2
Example: R = Z_15
Z(R)* = {3, 5, 6, 9, 10, 12}
Complete bi-partite graph
K_{2, 3}, girth = 4, diameter = 2
R is a commutative ring
Vertices = Z(R)* = { a in R\{0} such that ab =0 for some nonzero b in R}
Edges: if a, b are vertices, then a—b is an edge iff ab = 0
d(a, b) = length of shortest walk (path) from a to b.
diameter (graph) = sup{d(a, b) | a, b are distinct vertices}
girth(graph) = length of shortest cycle. If a graph has no cycles, then it has girth equals to infinity. *

4. Example 3
R = Z_30
Z(R)* = {2, 3, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28}
Diameter = Girth = 3

5. Results:
Anderson-Livingston (D.F. Anderson, P.S. Livingston, The zero- divisor graph of a commutative ring, J. Algebra 217 (1999) 434– 447):
1. Let R be a commutative ring. Then the zero-divisor graph of R is connected with diameter = 0, 1, 2, or 3 and with girth = 3, 4, 5, 6, 7, or infinity.
2. If Z(R) not equal {0}, then R is finite iff Z(R) is finite.
Mulay (S.B. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra 30 (7) (2002) 3533–3558):
Girth = 3, 4, or infinity

6. Results:
Thomas. G. Lucas, The diameter of a zero- divisor graph, J. Algebra 301(2006), 3533— 3558:
Diameter(zero-divisor graph) = 1 iff xy = 0 for every distinct x, y in Z(R).
Diameter(zero-divisor graph) = 2 iff (Nil(R) = {0}, |Z(R)*| >= 3, and R has exactly 2 minimal prime ideals ) OR ( Z(R) is an ideal of R such that Z(R)^2 not = {0} and ann{a, b} not = {0} for every distinct a, b in Z(R)*).
Diameter(zero-divisor graph) = 3 iff ann{a, b} = {0} for some distinct a, b in Z(R)* and either ( Nil(R) = {0} and R has at least 3 minimal prime ideals ) or ( Nil(R) not = {0}. If Nil(R) not = {0} and Z(R) is not an ideal of R, then diameter(zero-divisor graph) = 3.)
Nil(R) = {x in R | x^n = 0 for some positive integer n}
Observe Nil(R) is a subset of Z(R).
Ann{a, b} = {x in R | xa = 0 and xb = 0}
Diam( R = Z_2 X Z_2) = 1
Note Z(R) = {(0, 1), (1, 0)}
Diam( R = Z_3 X Z_7) = 2
Diam (R = Z_8) = 2
Diam(R = Z_3 X Z_5 X Z_7) = 3.
Diam (R = Z_3 X Z_4) = 3

7. Results:
Anderson-Mulay: D.F. Anderson and S. B. Mulay, S.B., On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra 210 (2) (2007),
Suppose Nil(R) = {0}. Girth(zero-divisor graph) = 4 iff T(R) is ring-isomorphic to F_1 X F_2, where F_1, F_2 are fields with |F_1|, |F_2| >= 3 iff zero-divisor graph of R is a complete-bi-partite graph.
(Here T(R) = R_S, where S = R\Z(R).)
Suppose Nil(R) not = {0}. Girth(zero- divisor graph) = 4 iff R is ring-isomorphic to B X D, where D is an integral domain with |D|>=3 and B = Z_4 or B = Z_2[X]/(X^2).
R is an integral domain if Z(R) = {0}.
Example: R = Z_15
Z(R)* = {3, 5, 6, 9, 10, 12}
complete bi-partite graph, K_{2, 3}, girth = 4, diameter = 2
Example: R = Z_{12} = Z_4 X Z_3
Z(R)* = {2,3,4,6,8,9,10}
Nil(R) not = {0}, Girth = 4, diameter = 3.

8. Suppose Nil(R) = {0}. Then girth(zero- divisor graph) = infinity iff T(R) is a ring- isomorphic B or to Z_2 X K, where K is a field. (again, T(R) = R_S, where S = R\Z(R.)
Suppose Nil(R) not = {0}. Then girth(zero- divisor of R) = infinity iff R is ring- isomorphic to B or Z_2 X B where B = Z_4 or Z_2[X]/(X^2) or graph(zero- divisor graph) is a star graph.
A. Badawi, “On the annihilator graph of a commutative ring,” Comm. Algebra, Vol.(42)(1), (2014). Star(zero- divisor graph): K_{1, 1}, K_{1, 2}, or
K_{1, infinity}
Results
Example R = Z_{10} =
Z_2 X Z_5 = T(R).
Z(R)* = { 2, 4, 5, 6, 8} 5
Girth = infinity, diameter = 2
Example R = Z_2 X Z_4
(1, 0) (1, 2)
(0, 1) (0, 2) (0, 3)
Girth = infinity, diameter = 3
OR OR
………. INFINITY

9. Planar, Genus 1 zero-divisor graph.
1. Richard Belshoff and Jeremy Chapman,
Planar zero-divisor graphs, J. Algebra 316 (2007) 471–480.
2.Cameron Wickham, Classification of rings with genus one zero-divisor graphs, Comm. Algebra 36 (2008), pages
David F. Anderson and Ayman Badawi, On the zero-divisor graph of a ring, Comm. Algebra 36(2008), We studied zero- divisor graph of phi-rings, R is called phi-ring if Nil(R) is a divided prime ideal of R (i.e., Nil(R) is prime and Nil(R) is a subset of aR for every a in R\Nil(R). In this paper, we showed that if Nil(R) \not = Z(R) and it is a divided prime, then Nil(R) must be infinite.

10. Zero-divisor graph for non-commutative rings
A. Akbari and A. Mohammadian, Zero-divisor graph of non- commutative rings, J. Algebra, 296 (2006) 462–479.
Frank DeMeyer and Lisa DeMeyer, Zero divisor graphs of semigroups, J. Algebra 283 (2005),

11. Annihilator graph of a commutative ring
A. Badawi, “On the annihilator graph of a commutative ring,” Comm. Algebra, Vol.(42)(1), (2014).

12. Annihilator graph in the sense of Badawi paper
Ann{ab} = {x in R | xab = 0 }
Ann{a} = {x in R | xa = 0}
Example: R = Z_8 2
Zero divisor graph of Z_8
Annihilator graph of Z_8
Complete Graph , Diameter = 1.
R is a commutative ring
Vertices = Z(R)* = { a in R\{0} such that ab =0 for some nonzero b in R}
Edges: if a, b are vertices, then a—b is an edge iff ann{ab} not = ann{a} U ann{b}
d(a, b) = length of shortest walk (path) from a to b.
diameter (graph) = sup{d(a, b) | a, b are distinct vertices}
girth(graph) = length of shortest cycle. If a graph has no cycles, then it has girth equals to infinity. *

13. Results on Annihilator graph
From now and on, ZG(R) denotes the zero- divisor graph of R and AG(R) denotes the annihilator graph of R.
AG(R) is connected and with diameter 1 or 2.
a--b is an edge in AG(R) iff ann{ab} not = Ann{a} and ann{ab} not = ann{b}
Suppose AG(R) not = ZG(R) and Nil(R) not = {0}. Then Girth(AG(R)) = 4 if and only if R is ring isomorphic to either Z_2 X Z_4 or
Z_2 X Z_2[X]/(X^2) .
Suppose AG(R) not = ZG(R) and Nil(R) = {0}. Then Girth(AG(R)) = 3.
Suppose AG(R) not = ZG(R). Then girth(AG(R)) = 3 or 4
R = Z_2 X Z_4
(1, 0) (1, 2)
(0, 1) (0, 2) (0, 3)
ZG(R),.
(1, 0) (1, 2)
(0, 1) (0, 2) (0, 3)
AG(R) = K_{2,3}
A complete bi-partite
graph

14. Results
Suppose AG(R) not = ZG(R). Then ZG(R) is a star graph iff ZG(R) = K_{1, 2}
Suppose AG(R) not = ZG(R). Then ZG(R) is a star graph iff AG(R) = K_3.
Suppose Nil(R) = {0}. Then AG(R) = ZG(R) if and only if R has exactly 2 minimal prime ideals
Suppose Nil(R) = {0}. Then AG(R) = ZG(R) if and only if girth(ZG(R)) = girth(AG(R)) = 4 or infinity.
Suppose Nil(R) = {0}. Then AG(R) is complete iff ZG(R) is complete iff R is ring-isomorphic
Z_2 X Z_2
Example :
R = Z_8 2
ZG(R)
R = Z_3 X Z
AG(R) = ZG(R)
R = Z X Z X Z_3
AG(R) NOT = ZG(R)

15. Results on Annihilator graph
Suppose Nil(R) \not = {0} and it is the only prime ideal of R. Then AG(R) is complete.
Suppose that Nil(R) Not = {0}. Then
Girth(AG(R)) = 4 iff AG(R) = K_{2, 3}
Iff R is ring-isomorphic to either Z_2 X Z_4 or Z_2 X Z_2[X]/(X^2) .
R = Z_{16}
Nil(R) = {2, 4, 6, 8, 10-, 12, 14} is the only prime ideal of R. AG(R) is complete and AG(R) Not = ZG(R)
Girth(ZG(R)) = Girth(AG(R)) = 3 .
Diameter (AG(R)) = 1, Diameter (ZG(R))) = 2.

16. Totol graph of commutative rings
Badawi and D.F. Anderson, ” The total graph of a commutative ring,”
J. Algebra 320 (2008),
Google scholar (79 citation), Math Science net (26 citation)

17. Total graph in the sense of Anderson Badawi-Paper
R is a commutative ring
Vertices = R , recall Z(R) = {a in R | ab = 0 for some nonzero b in R}
Edges: if a, b are vertices, then a—b is an edge iff a + b in Z(R)
d(a, b) = length of shortest walk (path) from a to b.
diameter (graph) = sup{d(a, b) | a, b are distinct vertices}
girth(graph) = length of shortest cycle. If a graph has no cycles, then it has girth equals to infinity.
R = Z_6 = {0, 1, 2, 3, 4, 5}
Z(R) = {0, 2, 3, 4} 5
TG(R) is connected,
Diameter = 2
Girth 3.

18. Total graph: Suppose Z(R) is an ideal of R. Then
TG(R) is disconnected.
Let m = |R|/ |Z(R)|, n = |Z(R)|. Then:
I) If 2 in Z(R) , then TG(R) is a union of m disjoint K_{n}.
Example R = Z_8, Z(R) = {0, 2, 4, 6}
TG(R) :
II) If 2 not in Z(R), then TG(R) is a union of one K_n and (m - 1)/2 disjoint complete bi- partite graphs K_{n, n}
R = Z_9 = {0, 1, …, 8}
Z(R) ={0, 3, 6} = n
complete
Complete bi-partite graph
K_{3, 3}
Note m = 9/3 = 3
Hence (3 – 1)/2 = 1

19. Total Graph: Suppose Z(R) is not an ideal of R. Then
See previous slide for an example
Suppose Z(R) is not an ideal of R. Then
TG(R) is connected if and only if
1 = z_1 + z_2 + … + z_n for some zero-divisors z_1, …, z_n in Z(R). Furthermore, if n is the smallest such integer, then diameter(TG(R)) = n.
3. Let Reg(R) be the induced sub-graph of
TG(R) with vertices R\ Z(R). If Reg(R) is connected, then TG(R) is connected.

20. Total graph
For each integer m > = 2, there is a ring R such that diameter(TG(R)) = m.
Let A be a commutative ring and M be an A-module, R = A (+) M is a commutative ring where (a_1, m_1) + (a_2, m_2)= (a_1 + a_2, m_1 + m_2) and(a_1, m_1)(a_2, m_2) = (a_1a_2, a_1m_2 + a_2m_1).
Take A = Z[X_1, …, X_{n-1}], K be the quotient field of A,
P_0 = (X_1 + …+ X_{m-1}), for each i where 1 <= i <= m- 2
P_i = (X_i), and P_{m-1} = (X_{m+1} + 1). Let F = the union of all these P_i’s. Let H = A \ F , S = A_H, and M = K/S. Now let R = A (+) M. One can check Z(R) = F (+)M and by construction diam(T(R)) = m.

21. Total Graph
Nothing to explain
R = Z_3 X Z_8 X Z_9.
TG(R) is connected with diameter 2
Let R be a commutative ring with 1 such that Z(R) is not an ideal of R. Then the total graph of R_S is always connected with diameter 2, where S = R \Z(R). Let R be a finite commutative ring such that Z(R) is not an ideal of R. Then the total graph of R is connected with diameter 2 If you are interested to know more: Download all related papers for free

22. References on total graph
D. F. Anderson and A. Badawi, The total graph of a commutative ring without the zero element, J. Algebra Appl. 11, No. 4 (2012) (18 pages).
.D. F. Anderson and A. Badawi , “The generalized total graph of a commutative ring,” Journal of Algebra and Its Applications (JAA), .Vol. 12, No. 5 (2013).
S. Akbari, M. Jamaali and S. A. Seged Fakhari, The clique numbers of regular graphs of matrix algebras are finite, Linear Algebra Appl (2009) 1715–1718.
[2] S. Akbari, D. Kiani, F. Mohammadi and S. Moradi, The total graph and regular graph of a commutative ring, J. Pure Appl. Algebra 213 (2009) 2224–2228.

23. References on Total Graph
H. R. Maimani, C. Wickham, and S. Yassemi, Rings whose total graphs have genus at most one, Rocky Mountain J. Math., to appear.
N. Ashra, H. R. Maimani, M. R. Pournaki, and S. Yassemi, Unit graphs associated with rings, Comm. Algebra 38(2010),
Z. Barati, K. Khashyarmanesh, F. Mohammadi, and Kh. Nafar, On the associated graphs to a commutative ring, J. Algebra Appl. 11, No. 2 (2012) (17 pages).