[PPT] - A Refinement of Cayley Graphs Associated to A. R. Naghipour Rings PowerPoint Presentation

SLIDE 1

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

A Refinement of Cayley Graphs Associated to Rings

A. R. Naghipour

Shahrekord University, Iran. 5 March 2018 Discrete Maths Research Group.

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SLIDE 2

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Outline

1 Introduction 2 A Refinement of Cayley Graphs Associated to Rings

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SLIDE 3

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Introduction

Some notations and structure for commutative rings Examples of commutative rings (1) Zn = Z/nZ. (2) Fn= field of order n. Ideals and Maximal ideals Let R be a commutative Ring with identity. (1) An ideal in R is an additive subgroup I ⊆ R such that Ix ⊆ I for all x ∈ R. (2)I is called maximal ideal if there is no ideal J with I J R.

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SLIDE 4

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Introduction

Some notations and structure for commutative rings Examples of commutative rings (1) Zn = Z/nZ. (2) Fn= field of order n. Ideals and Maximal ideals Let R be a commutative Ring with identity. (1) An ideal in R is an additive subgroup I ⊆ R such that Ix ⊆ I for all x ∈ R. (2)I is called maximal ideal if there is no ideal J with I J R.

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SLIDE 5

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Introduction

Some notations and structure for commutative rings Examples of commutative rings (1) Zn = Z/nZ. (2) Fn= field of order n. Ideals and Maximal ideals Let R be a commutative Ring with identity. (1) An ideal in R is an additive subgroup I ⊆ R such that Ix ⊆ I for all x ∈ R. (2)I is called maximal ideal if there is no ideal J with I J R.

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SLIDE 6

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Local rings

Definition of local ring Call a ring R local if R has exactly one maximal ideal. Examples of local rings (1) Z4. (2) Z9. (3) Zp2, where p is a prime number. (4) Zp[X]

(X2) , where p is a prime number.

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SLIDE 7

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Local rings ( Z4 and Z9)

4

{0,1,2,3}  0,2 1,3

9

{0,1,2,...,8}  0,3,6 1 2 4 5 7 8

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SLIDE 8

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Structure of finite commutative rings

(1) Let R be a finite commutative ring. Then R = R1 × R2 × · · · × Rk, where Ri is a local ring. (2) Let R be a finite commutative ring. Then R/J(R) = F1 × F2 × · · · × Fk, where Fi is a Field. (Here J(R), the Jacobson radical of R, is the intersection of all maximal ideals of R)

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A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Some facts about Jacobson radical

Structure of Jacobson radical J(R) = {r ∈ R|1 + rx is unit for all x ∈ R}. Theorem u is a unit in R if and only if u + J(R) is a unit in R/J(R).

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SLIDE 10

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Some important graphs associated to rings (1) Zero divisor Graph of a ring. (2) Cayley Garph of a ring. (3) Unit Graph of a ring. (1) Zero divisor graph of a ring The concept of a zero-divisor graph of a commutative ring was first introduced by Beck. (In his work all elements of the ring were vertices of the graph). V (Γ(R) = Z(R) \ {0} and two distinct vertices x and y are adjacent if and and only if xy = 0. [Beck] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), 208-226.

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A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Some examples of Zero divisor graphs

2 4 2

[ ]

r (

) x x

3 9 2 2 2

[ ] ,

r (

) x x 

2 6 8 3

[ ] ,

r (

) x x

2 4 2 2 2

[ , ] [ ]

r

( , , ) ( ) x y x x xy y x

3 3



5 25 2

[ ]

r (

) x x

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SLIDE 12

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Some examples of Zero divisor Graphs

2 4



2 7



3 5



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SLIDE 13

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

(2) Cayley graph of a ring The Cayley graph Γ(R) is the graph with vertex set R such that two distinct vertices x and y are adjacent if and only if x − y is unit in R. Unitary Cayley graphs are introduced in: Lucchini, et al.

A. Lucchini, A. Maroti, Some results and questions related to

the generating graph of a finite group, Proceedings of the Ischia Group Theory Conference, 2008. Akhtar, et al.

R. Akhtar, M. Boggess, T. Jackson-Henderson, I. Jim´enez, R.

Karpman, A. Kinzel and D. Pritikin, On the unitary Cayley graph of a finite ring, Electron. J. Combin. 16 (2009) #R117.

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SLIDE 14

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

A general example of Cayley graph

(R, M) is a local ring, R/M = {x1 + M, x2 + M, . . . , /xt + M} and |xi + M| = ni| for all 1 ≤ i ≤ t.

1

n

K

2

n

K

3

n

K

i

n

K

t

n

K

1

x M 

2

x M 

3

x M 

i

n

x M 

t

n

x M 

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SLIDE 15

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

(3) Unit graph of a ring The unit graph Γ(R) is the graph with vertex set R such that two distinct vertices x and y are adjacent if and only if x + y is unit in R. The unit graphs are introduced in: Fuchs

E. Fuchs, Longest induced cycles in circulant graphs, Electron.
J. Combin. 14 (2005) #R52.

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SLIDE 16

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Some examples unit graphs

3



4



2 2



 

6 2 3

    

3 3

  

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SLIDE 17

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Definition in this talk A Refinement of Cayley Graphs Associated to Rings Let R be a finite ring and U(R) be the set of all unit elements

f R. The Unit graph Γ(R) is the graph with vertex set R such

that two distinct vertices x and y are adjacent if and only if there exists a unit element u of R such that x + uy is unit in R. If we omit the word ”distinct”, we obtain the graph Γℓ(R); this graph may have loops.

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SLIDE 18

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Motivation (1) The study of algebraic structures using the properties of graphs, (2) Some result about unit 1-stable range rings. We recall that a ring R is said to have unit 1-stable range if, whenever Rx + Ry = R, there exists u ∈ U(R) such that x + uy ∈ U(R).

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SLIDE 19

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Some examples

2 2

( ( ) )   

 



3

( )  

2 3 2 3

( ( ) )     

 

   1 1 2 1 2 3 4 5 6 7 8 1 3 2

4 4

( ( ) )   

 



9

( )   (0,0) (1,1) (0,1) (1,0) (0,2) (1,2)

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SLIDE 20

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Outline

1 Introduction 2 A Refinement of Cayley Graphs Associated to Rings

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SLIDE 21

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Some Properties of Γ(R)

When Γ(R) is a Complete Bipartite Graph Let R be a ring and let M be a maximal ideal of R such that |R/M| = 2. Then Γ(R) is a complete bipartite graph if and

nly if R is a local ring.

Degree of Vertices Let R be a local ring with maximal ideal M such that |R/M| > 2 and let x ∈ R. Then deg(x) = |R| − 1 if x ∈ U(R), |U(R)|

therwise.

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SLIDE 22

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Some Properties of Γ(R)

Connectedness (1) Γ(R) is connected if and only if R/J(R) has at most one Z2 as a summand. Connectedness (2) Γ(R) is not connected if and only if R = Z2 × Z2 × R3 × · · · × Rn. (Ri is a local ring)

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SLIDE 23

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Some Properties of Γ(R)

Clique and Chromatic number Let R = R1 × R2 × · · · × Rn be a ring, where Ri is a local ring with maximal ideal Mi. Then χ(Γ(R)) = ω(Γ(R)) = 2 if |Ri/Mi| = 2 for some i |U(R)| + n

therwise.

Proof Let |Ri/Mi| = 2, for some 1 ≤ i ≤ n. Then M := R1 × · · · × Ri−1 × Mi × Ri+1 × · · · × Rn is a maximal ideal of R such that |R/M| = 2. Therefore ω(Γ(R)) = 2. Now suppose that |Ri/Mi| > 2 for all 1 ≤ i ≤ n. We set: Si := U(R1) × · · · × U(Ri−1) × Mi × Ri+1 × · · · × Rn. Sn+1 := U(R1) × U(R2) × · · · × U(Rn).

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SLIDE 24

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Some Properties of Γ(R)

Proof It is easy to see that Si Sj = ∅, for all i = j, and i=n+1

i=1

Si = R. Sn+1 is a clique. Set C := Sn+1 ∪ {(0, 1, 1, . . . , 1), (1, 0, 1, . . . , 1), (1, 1, . . . , 1, 0)}. It is easy to see that C is a clique of Γ(R). Since Si (1 ≤ i ≤ n) is a coclique, then every arbitrary clique of Γ(R) contains at most one element of Si (1 ≤ i ≤ n). Therefore ω(Γ(R)) = |U(R1)|×|U(R2)|×· · ·×|U(Rn)|+n = |U(R)|+n. This argument also shows that χ(Γ(R)) = |U(R1)|×|U(R2)|×· · ·×|U(Rn)|+n = |U(R)|+n.

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A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Some Properties of Γ(R)

Vertex-connectivity and Edge-connectivity Let κ(G) and κ′(G) denote the vertex-connectivity and edge-connectivity of a graph G, Theorem κ(Γ(R)) = κ′(Γ(R)) = |U(R)| = δ(Γ(R)).

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A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Some Properties of Γ(R)

Hamiltonian cycle Let R be a ring such that R = Z2. Then Γ(R) is a connected graph if and only if Γ(R) is Hamiltonian. Matching Γ(R) has perfect matching if and only if |R| is an even number.

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A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Some properties of Γ(R)

Planarity for Γ(R) Γ(R) is planar if and only if R is one of the following rings (1) Z2 × Z2 × · · · × Z2

n times

. (2) Z2 × Z2 × · · · × Z2

n times

×Z4. (3) Z2 × Z2 × · · · × Z2

n times

×F4. (4) Z2 × Z2 × · · · × Z2

n times

× Z2[x]

(x2) .

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SLIDE 28

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Some Properties of Γ(R)

Connectedness Let R = R1 × R2 × · · · × Rn be a ring, where Ri is a local ring with maximal ideal Mi. Then Γ(R) is connected if and only if R satisfies one of the following: (1) |R1/M1| = |R2/M2| = 2 and n ≥ 2. (2) |R1/M1| = 2 and |Rk/Mk| > 2 for every 2 ≤ k ≤ n. Hamiltonian Cycle Let R = R1 × R2 × · · · × Rn be a ring, where Ri is a local ring with maximal ideal Mi. Then Γ(R) is Hamiltonian if and only if R satisfies any one of the following three cases: (1) |R1/M1| = |R2/M2| = 2 and n ≥ 2. (2) n = 2 with |R1/M1| = 2, |R2/M2| > 2 and at least one of R1 or R2 is not field. (3) n ≥ 3, |R2/M2| = 2 and |Rk/Mk| > 2 for every 2 ≤ k ≤ n.

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A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Some Properties of Γ(R)

Planarity for Γ(R) Γ(R) is planar if and only if R is one of the following rings (1) Z2, Z4, Z2[x]

(x2) , Z8, Z2[x] (x3) , Z4[x] (2x,x2−2), Z4[x] (x,2)2 , Z4[x,y] (x,y)2 .

(2) F is a field with |F| > 2, F4[x]

(x2) , Z4[x] (1+x+x2), Z9, Z3[x] (x2) .

(3) Z2 × Z2, Z2 × Z3, Z2 × F4, Z3 × Z3, Z3 × F4, F4 × F4.

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SLIDE 30

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Used papers for this talk [1] A. R. Naghipour, M. Rezagholibeigi. A Refinement of the Unit and Unitary Cayley Graphs of a Finite Ring. Bull. Korean

Math. Soc. 53 (2016), No. 4, pp. 1197–1211.

[2] T. Tamizh Chelvam and S. Anukumar Kathirve, On generalized unit and unitary Cayley graphs of finite rings, Journal of Algebra and its Applications. Accepted, 2018. [3] A. R. Naghipour, M. Rezagholibeigi. A Refinement of the Unit and Unitary Cayley Graphs of a Noncommutative Ring, in preparation.

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SLIDE 31

A Refinement

f Cayley

Graphs Associated to Rings

A. R.

Naghipour Shahrekord University, Iran. Introduction A Refinement

f Cayley

Graphs Associated to Rings

Acknowlegement

Thank you for your attention

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