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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
P. Srinivas
H.No 6-8-25, Ravindra Nagar, Nalgonda
Sri Venkateswara Engineering College
Suryapet, Nalgonda, A.P, India
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.
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 ⁄ }
46 M. Vijay Kumar and P. Srinivas
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
Irregular graph for lattice 47
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.
48 M. Vijay Kumar and P. Srinivas
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.
Irregular graph for lattice 49
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.
50 M. Vijay Kumar and P. Srinivas
Example: for the Graph
( ) { ( ) { } { } { } { } { } { }} Since
( ({ })) ({ }) { } ( ({ })) ({ })
{ } ( ({ })) ({ }) { }
Similarly { } { } { } ( )
Fig: 8
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Irregular graph for lattice 51
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Received: October, 2011