irregular graph for lattice -...

Pure Mathematical Sciences, Vol. 1, 2012, no. 1, 43 - 51

Irregular Graph for Lattice

M. Vijay Kumar

3 – 10 – 80, Reddy colony

Vagdevi P.G College

Warangal, A.P, India

[email protected]

P. Srinivas

H.No 6-8-25, Ravindra Nagar, Nalgonda

Sri Venkateswara Engineering College

Suryapet, Nalgonda, A.P, India

[email protected]

Abstract

In this paper by using the lattice condition of irregular graphs and classify these

graphs as orthographs, Boolean graphs, known and unknown results etc.

Mathematics Subject Classification: 05C

Keywords: Regular Graphs, Irregular Graphs, Neighborly Irregular Graphs, Ortho

Graphs and Boolean Graphs

Introduction

R. Balakrishnan and K. Ranganathan were follow the notation and terminology

for considering finite simple connected graphs. Any non – trivial graphs must have at

least two vertices of the same degree. [4, 10] studied exactly two vertices of the same

degree. A graph which contains at least one vertex of each possible degree is known

planar graph and there exists a unique of order n.

mailto:[email protected]

mailto:[email protected]

44 M. Vijay Kumar and P. Srinivas

1. Preliminaries

I. Lattice of regular Graphs

Definition 1: If the graph with vertex set{ } and edge set

{ ⌊

⌋ } degree of each vertex and it is denoted .

( ) { ⌊

⌋

⌊

⌋

For example shown in Figure 1

Fig: 1

Definition 2:

A connected graph G is called highly irregular if every vertex of G is adjacent

only to vertices with distinct degrees [1] Therefore two vertices U and V of G are both

adjacent to a vertex W of G then d(U) ≠ d(V) in G.

Example:

Fig: 2 (Highly regular)

Irregular graph for lattice 45

Definition 3:

The highly irregular bipartite graph [12] with bipartite graph with bipartite sets

{ } And { } and edge set

{ ⁄ }

i.e. ( ) ( ) For Example

K 23

K 44

Fig: 3

[5] Introduced Lattice theory we have ortho lattices, ortho modular Lattices, modular

ortho Lattices and Boolean algebra.

Definition 4:

An ortho lattice is a bounded lattice L equipped with a function c: L→ L

Satisfying

1. ( ) ( )

2. ( ( )) 3. ( ) For all x and y in L.

Note:

1. A function satisfying condition (1) is called Antitone.

2. C is called ortho complementation of the ortho lattice.

3. A complete ortho lattice is an ortho lattice which is a complete lattice.

Definition 5 [13]:

A Graph G construct a complete ortho lattice L(G), called the neighborhood ortho

lattice of G.

Let 2G

denote the lattice of all subsets of the vertex set of G.

Define ( ) { ( ) a ⁄ }


Remark: A subset A of V(G) is said to be closed if ( ( ))

For example consider the Graph given in Figure 4.

Fig : 4

Here ({ }) { } ( ({ })) ({ }) { } Hence { } ( ) similarly

{ } { } ( )

Now ( ({ })) ({ }) { } { } ( ({ })) ({ }) { } { }

Therefore { } { } ( ) Thus L(G) = {Ǿ, {a}, {b}, {c}, {a, b}, {b, c}, {c, a}, {a, b, c} = V(G), }. The Hasse

diagram of the underlying lattice of the neighborhood ortho lattice is shown in Figure 5.

Fig: 5


A graph G is said to be an ortho modular graph if the lattice of G, namely L(G), is

an ortho modular lattice. A graph G is said to be a modular ortho graph if the lattice of G

is a modular ortho lattice. A graph G is a Boolean graph if the lattice of G is a Boolean

algebra. As every graph gives rise to an ortho lattice, we call every graph as an ortho

graph [8]. It has been proved [7] that an ortho lattice is ortho modular if and only if it has

no isomorphic copy of O6, where O6 is the lattice given in Figure 6.

Fig 6

Lattice approach to irregular graphs and and we classify these graphs as a Boolean

graphs and ortho graphs respectively Therefore ( ( ))

( ) { ( ) a a Consider the following Example

( ) { }

2. Lattice of Irregular Graphs

Case I: Let are adjacent vertices

Therefore ({ }) { } ( { }) { } { } Similarly ({ }) { } ( { }) { } { }

Case II: { } { } { } { ( ){ } { }} = Isomorphic to and atom 2

= is Boolean Graph.

Case III: Graph ( ) { ( ){ } { }} Here { } { }

{ { }} { } { }

The lattice diagram ( ) is isomorphic to so is Boolean graph.


Case IV: ( ) { ( ){ } { } { } { } { } { }}

= Isomorphic to and atom 3

= is Boolean Graph.

Case V: ( ) { ( ) { }{ } { } { } { } { }} And

the Corresponding Lattice Diagram

Here ( ) ( ) all Above Cases figure as shown below

Fig: 7

Theorem:

Lattice of In is finite Boolean algebra, is isomorphic to a power set Boolean

algebra. Specifically to the set Boolean algebra of the set all its atoms and In is Boolean

Graph.

Proof: Let (X,+, .)Be a Boolean algebra. Suppose | | then X is isomorphic to the

power set Boolean algebra of the empty set.

Assume | | then and so X had at least one atom.

Let be the distinct atoms of X Now Let S be the set {1, 2 …n} we asset that the power set Boolean algebra

( ( ) ) is isomorphic to ( ) Define ( ) As follows,

Say In of P(S) is some subset of {1,2,3,……..n}.

We let ( ) ∑ ( ) In other words

( ) Is the sum of atoms whose indices are in . Clearly ( ) ( ) Every element of X can uniquely expressed as a sum of atoms specifically if then x

is the sum of all atoms contained in x. the function f is a bijection

In order to show that is an isomorphism of two Boolean algebras.


Specifically for any to subsets I and J of S we must show

1. ( ) ( ) ( ) 2. ( ) ( ) ( ) 3. ( ) ( )

We verify these conditions one-by-one

For Notational convenience Let

{ } And

{ } Where p, q, r are non - negative integer S

and

And

Then { } And

{ } Now for i ( ) ∑

∑

∑

∑

∑

∑

∑

( ) ( ) (By law of tautology)

For i i

( ) ( ) (∑

∑

) (∑

∑

)

∑ [Every two distinct atoms of X are mutually disjoint]

∑ (By law of tautology)

( ) And Finally

( ) ( ) ( ( ) ( ) And

( ) ( ) ( ( ) ( ) So uniqueness of complements ( ) [ ( )]| iii & Completes the proof that is an Isomorphism.

So X is Isomorphic to P(S) but obviously P(S) is isomorphic to the power set Boolean

algebra of the set { }

Corollary: If X is a finite Boolean algebra to Boolean graph then | | for some non

– negative integer n.

Proof: the power set of a set with n elements have cardinality . (n be the number of

atoms in X).

Theorem:Highly irregular bipartite Graph is an Ortho Graph (since lattice is an ortho lattice.


Example: for the Graph

( ) { ( ) { } { } { } { } { } { }} Since

( ({ })) ({ }) { } ( ({ })) ({ })

{ } ( ({ })) ({ }) { }

Similarly { } { } { } ( )

Fig: 8

REFERENCES

1. Ebrahim Salehi, On-degree of Graphs, JCMCC 62 (2007), pp. 45 – 51.

2. E. K. R. Nagarajan and M. S. Mutharasu, A Study on Orthographs, Ultra Science,

17(3)M, 409 – 18 (2005).

3. E. K. R. Nagarajan and M. S. Mutharasu, “Modular Orthographs”, Ultra Science,

18(2)M, 293– 98 (2006).

4. Gudrun Kalmbach, Orthomodular Lattices, Academic Press, London, (1983).

5. G. Birkhoff, Lattice Theory, 3rd ed., Amer. Math. Soc., Providence, R.I., 1967.

6. James W. Walker, “From Graphs to Ortholattices and Equivariant Maps”, Journal

of combinatorial Theory, Series B 35. 171 – 192 (1983).

7. L. Nebesky, On Connected Graphs Containing Exactly Two Points of the Same

Degree, Casopis Pro Pestovani Matematiky, 98 (1973), 305–306.

8. M. Behzad and G. Chartrand, No Graph is Perfect, Amer. Math. Monthly 74

(1967), 962 – 963.

9. R. Balakrishnan and K. Ranganathan, A Text Book of Graph Theory, Springer

Verlag, (2000).


10. Selvam, “Highly Irregular Bipartite Graphs”, Indian Journal of Pure and

Applied Mathematics, 27(6), 527-536, June 1996.

11. Selvam Avadayappan, P. Santhi and R. Sridevi, “Some Results on Neighborly

Irregular Graphs”, International Journal of Acta Ciencia Indica, vol. XXXII M,

No. 3, 1007– 1012, (2006).

12. S. Gnaana Bhragsam and S. K. yyaswamy, “Neighborly Irregular Graphs”,

Indian Journal of Pure and Applied Mathematics, 35(3): 389-399, March 2004.

13. Yousef Alva, Gary Chartrand, F. R. K. Chung, Paul ErdÖs, R. L. Graham and O.

R. Oellermann, “Highly Irregular Graphs”, Journal of Graph Theory, 11 (1987),

235-249.

Received: October, 2011

irregular graph for lattice -...

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