Generating Topology on Graphs by Operations on Graphs

Applied Mathematical Sciences, Vol. 9, 2015, no. 57, 2843 - 2857

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2015.5154

Generating Topology on Graphs by

Operations on Graphs

M. Shokry

Physics and Engineering Mathematics Department, Faculty of Engineering

Tanta University, Egypt, Tanta, Zip code 3111, Tanta, Egypt

Copyright © 2015 M. Shokry. This is an open access article distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in

any medium, provided the original work is properly cited.

Abstract

The focus of this article is on various approaches to discerning topological

properties on a connected graph by using M-contraction, D-Deletion

neighborhoods. We introduce a new definition of neighborhood which is built on

the choice of the distance between two vertices. A comparison between these

types of results from a new formed topologies and neighborhoods is discussed.

Also we discussed the containment properties and compared the number of

elements in the sets of these neighborhoods including closed sets and open sets.

And we have strengthened that by the vital examples.

Keywords: Graph theory, Rough set, Topology, Fuzzy set and Data mining

1. Preliminaries

Topological structures are mathematical models, which are used in the

analysis of data on which the notion of distance is not available. We believe that

topological structures are important modification for knowledge extraction and

processing [5]. Some of the basic concepts in topology which are useful for our

study are given in this paper. Graphs are some of the most important structures in

discrete mathematics [1]. Their ubiquity can be attributed to two observations.

First, from a theoretical perspective, graphs are mathematically elegant. Even

though a graph is a simple structure, consisting only of a set of vertices and a

relation between pairs of vertices, graph theory is a rich and varied subject. This is

partly due the fact that, in addition to being relational structures, graphs can also

be seen as topological spaces, combinatorial objects [1], and many other

mathematical structures. This leads to the second observation regarding the

2844 M. Shokry

importance of graphs, many concepts can be abstractly represented by graphs[6],

making them very useful from a practical viewpoint.

A Graph G is an ordered pair of disjoint sets ( V,E ) where V is nonempty

set and E is a subset of unordered pairs of V. The vertices and edges of a graph G

are the elements of V=V(G) and E=E(G) respectively . We say that a graph G is

finite (resp. infinite) if the set V(G) is finite ( resp. infinite ) . The degree of a

vertex u ϵ V(G) is the number of edges containing u . If there is no edge in a graph

G contains a vertex u, then u is called an isolated point, and so the degree of u is

zero, [6].

A graph that is in one piece, so that any two vertices are connected by a

path, is a connected graph, and disconnected otherwise. Clearly any disconnected

graph G can be expressed as the union of connected graphs, each of which is a

component of G, [1], [6].

A topological space (X, τ) is disconnected space, if there are two nonempty

disjoint open sets A and B, such that X = A U B. Otherwise, X is connected space,

[3], [4].

Some applications used generalized topological spaces derived from a

graph. We asked about if the structure of given phenomena depends on more than

sub graphs. So, the main problem here is devoted to answer the following

questions:

1- How can we define a (generalized) topological space by using an arbitrary

family of different neighborhoods of vertices?

2- Is there a correspondence between a certain generalized topological space and

neighborhoods of vertices?

In this paper we discuss a new method to generate topology τ on graph by

using new method of taking neighborhood is determining two fixed vertices on

the graph and calculate each vertex and its incident edge which away from the two

fixed vertices according to the degree of distance of each one of them . Each edge

and vertices as every open set in topology contained vertex and its incident edge,

also every set of singleton edge is open set.

We discuss topological concept on graph such that construct topology on

special cases in a graph like comb graph, ladder graph and skeletal graph. We can

apply this method in determining the distance between two vertices in a graph of

airline connections is the minimum number of flights required to travel between

two cities.

Let G= (V, E) be a graph with diameter d [1, 6, 7], and let V (G) and E (G)

denote the vertex set and the edge set of X, respectively. For u, v ∈ V (X), we let

dG (u, v) (in short d (u, v)) denote the minimal path-length distance between u

and v.

Generating topology on graphs by operations on graphs 2845

We say that G is distance balanced if

|𝑣𝑘 ∈ 𝑉(𝐺): 𝑑(𝑣𝑘, 𝑢) ≤ 𝑑 (𝑣𝑘, 𝑣)| = |𝑣𝑘 ∈ 𝑉(𝐺): 𝑑(𝑣𝑘, 𝑣) ≤ 𝑑 (𝑣𝑘, 𝑢)| holds for an arbitrary pair of adjacent vertices u and v of G. Let uv be an

arbitrary edge of G. For any two integers i, j, we let

Bij(a, u) = {𝑣𝑘 ∈ 𝑉(𝐺): 𝑑(𝑣𝑘, 𝑎) = 𝑗 𝑎𝑛𝑑 𝑑 (𝑣𝑘, 𝑢) = 𝑖}

The sets Bij(a, u) give rise to a distance partitions of V(G) with respect to

the edge eau ∈ E(G) We say that X is strongly distance-balanced if |Bi−1i (a, u)| =

|Bii−1(a, u)|

2 Generating Topology by Contraction Edge Operations in Graph

In this section .We obtained a new result by some operations of graph like

M-Contraction edge and converted them to topological properties [5]. We will

clarify the method that how each and every one of these operations represents on

the graph. After applying these operations on specific forms of graph. We can

apply these methods in applications such that formation of maps or in knowing

the roads and planning the shortcuts roads between cities. Also removing the

destroyed roads or unfit for use between regions which don't affect the traffic

plan.

We introduce new topological method and definitions based on some

graph operations. Let G=(V,E) be a graph , subdivision of G is informally any

graph obtained from G by subdivision for some edges of G by drawing a new

paths between their ends, so that none of these paths has an inner vertex in V(G) .

We formed a new defined for neighborhoods of two fixed vertices Nij(a,u) ,

which contained each vertex and its incident edge which linked to two fixed

vertices according to their distance. The topology built by Nij(a,u) as a set of

subbase confirms some important topological properties between locations of all

vertices and edges with this neighborhoods.

Definition 2.1

Let G=(V,E) be a graph and 𝐻𝑖𝑗 ⊆ 𝐺 a subgraph generated by all paths with

length j from vertex a and length i from u

Nij(a, u) = {𝑣𝑘, 𝑒𝑘 : 𝑣𝑘 ∈ 𝑉(𝐻𝑖𝑗), 𝑒𝑘 ∈ 𝐸(𝐻𝑖𝑗), 𝐻𝑖𝑗 ⊆ 𝐺 , 𝑑(𝑣𝑘, 𝑢) ≤ 𝑖 , 𝑑 (𝑣𝑘, 𝑎)

≤ 𝑗 }

2846 M. Shokry

Example 2.1

Let G= (V, E) be a comb – graph

Fig (2.1)

Firstly, we evaluate the neighborhood of the two fixed vertices v(a) and v(u) :

N11 (a, u) = {{b, e1}, {d, e3}}

N21 (a, u) = {{b, e1 },{d , e5 , f, e3 } , {b , e2 , d, e3 }}

N12 (a, u) = {{b, e1 , c , e4 },{b , e1 , d , e2 } , {d , e3}}

N22 (a, u) = {{b, e1 , c , e4 },{b , e1 , d , e2 } , {d , e5 , f, e3 } , {b , e2 , d, e3 }}

N31 (a, u) = {{b, e1}, {d, e3, b, e2, c, e4}}

N13 (a, u) = {b, e1, d, e2, f, e5},{d , e3}}

N32 (a, u) = {{b, e1 , c , e4 },{b , e1 , d , e2 } ,{d , e3 , b, e2 , c , e4 }}

N23 (a, u) = {{b, e1 , d , e2 , f ,e5 } , { d , e5 , f, e3 } , {b , e2 , d, e3 }}

N33 (a, u) = {{b, e1 , d , e2 , f ,e5 },{d , e3 , b, e2 , c , e4}}

The set of basis

β = {{e1},{e2} ,{e3} ,{e4 } ,{e5 } , {b} , {d } , {b, e1 },{d , e3} ,{d , e5 , f, e3 } ,b, e2 ,

d, e3}, {b , e1 , c , e4 },{b , e1 , d , e2 },{d , e3 , b, e2 , c , e4 },{b , e1 , d , e2 , f ,e5} , {d ,

b , e2} ,

{d, f, e5}, {b, c ,e4}}

τ= {{ X , ∅, { e1 },{ e2 } ,{ e3} ,{e4} ,{e5} , {b} , {d },{b , e1 },{d , e3} ,{d , e5 , f,

e3 },

{b , e2 , d, e3}, {b , e1 , c , e4 },{ b , e1 , d , e2 },{d , e3 , b, e2 , c , e4 },{ b , e1 , d , e2 , f

,e5} ,

{d , b , e2} , {d , f, e5}, {b ,c ,e4} , { e1 , e2}, {e1 , e3} ,{e1 , e4} ,{e1 ,e5} ,{d , e3 , e1}

,

{d , e5 , f, e3 , e1 }, { b , e2 , d, e3 , e1}, {d , e3 , b, e2 , c , e4 , e1 }, {d , f, e5 , e1}, { e2 ,

e3} ,

{ e2 , e4} ,{ e2 , e5},{ e2 , b}, {e2 , d} , {b , e1, e2 },{d , e3 , e2} ,{d , e5 , f, e3, e2 } ,

{b , e1 , c , e4 , e2 }, {d , f, e5 , e2}, {b , e2 , c , e4 } , { e3 , e4 }, {e3 , e5 } ,{ e3 , b} ,{b ,

e1, e3 }, { b, e3 , c , e4 , e1 } , { b , e1 , d , e2, e3 }, { b , e1 , d , e2 , f ,e5 , e3} , {d , b , e2,

e3} , {d , f, e5 , e3}, { e4 , e5 },{ b , e4} ,{ c , e4 } .

e5 e4

c f

e2 a u b d e1 e3


Let G= (V, E) be a graph and e = xy an edge of a graph G = (V, E). The

contraction graph G/e obtained from G by contracting the edge e into a new vertex

Ve, which becomes adjacent to all the former neighbors of x and of y. Formally,

G/e=(V',E') where

V'= (V ∖{x, y})∪{Ve }(where Ve is the ‘new’ vertex, i.e. Ve ∉{V∪E}

E' = { {uw E |{ 𝑣, 𝑤} ∩ { 𝑥, 𝑦} = ∅} ∪ {𝑣𝑒 𝑤 ∶ 𝑥 𝑤 ∈ 𝐸 ∖ {𝑒} 𝑜𝑟 𝑦𝑤 ∈ 𝐸 ∖ {𝑒}}

Fig (2.2)

Definition 2.3

Let G= (V, E) be a graph and 𝐻𝑖𝑗 ⊆ G/e a subgraph generated by all paths

with length j from vertex a and length i from u in G with contractible edge e, the

M-contractible neighborhood is defined as

Mij(a, u) = {𝑣𝑘 , 𝑒𝑘 : 𝑣𝑘 ∈ 𝑉(𝐻𝑖𝑗), 𝑒𝑘 ∈ 𝐸(𝐻𝑖𝑗), 𝐻𝑖𝑗 ⊆ G/e , 𝑑(𝑣𝑘, 𝑢) ≤

𝑖 , 𝑑 (𝑣𝑘, 𝑎) ≤ 𝑗 }

We studied some topological concepts in generalized topological spaces

and extend some results to certain generalized topological space. These extensions

of some results presented for main reasons to show that not all topological spaces

can be formed through graph operation and specified some properties on graph so,

the main aim in this work was the methodology of obtained a link between graph

theory and topology concepts.

2848 M. Shokry

Example 2.2

Consider the following graph

Fig (2.3)

After evaluating the neighborhood of the two fixed vertices and construct the

topological space on it by used {∅, (M𝑖𝑗)

𝑘(a, u)} as set of basis. We will begin to

apply two operations (M- contraction edge) on it. Firstly, we notice from the

previous figure that c contract to b, such as {c, b} represent as b

(M11)1( (a , u) = {{b , e1 },{d , e3}}

(M21)1( (a , u) = {{b , e1 },{d , e5 , f, e3 } , {b , e2 , d, e3 }}

(M12)1( (a , u) = {{b , e1 , d , e2 } , {d , e3}}

(M22)1((a , u) = {{b , e1 , d , e2 } , {d , e5 , f, e3 } , {b , e2 , d, e3 }}

(M13)1( (a , u) = {b , e1 , d , e2 , f ,e5 },{d , e3}}

(M23)1( (a , u) = {{b , e1 , d , e2 , f ,e5 } , { d , e5 , f, e3 } , {b , e2 , d, e3 }}

The set of basis β ={{e1 },{e2 },{e3},{e5},{b},{d },{b,e1},{d,e3},{d ,e5 ,f, e3},

{b,e2 ,d,e3},{b,e1 , d ,e2 },{b,e1 , d, e2, f ,e5} , {d , b , e2} , {d , f,

e5}}. Secondly, we notice from the previous figure that f contract to d, such as {f,

d} represent as d

Fig (2.4)


(M11)2 (a, u) = {{b , e1 },{d , e3}}, (M2

1)2 (a , u) = {{b , e1 } , {b , e2 , d, e3 }}

(M12)2 (a, u) = {{b , e1 , d , e2 }, {d , e3}},(M2

2)2(a , u) = {{b , e1 , d,e2},{b , e2 , d,

e3 }}

The set of basis β = {{e1 },{e2},{e3},{b},{d },{b,e1},{d,e3},{b,e2 ,d,e3},{b, e1 ,d

,e2},{d,b,e2}}.

Fig (2.5)

Thirdly, We notice from the previous figure that b contract to d , such as {b , d}

represent as b (M11)3 (a , u) = {{b , e1 }}.The set of basisβ = {{e1 } ,{e3}, {b,e1} }.

Finally, it's clear from the previous example after applied edge contraction

on the graph, we found that the graph is connected, also we notice that the result

neighborhood of each step is (M𝑖𝑗)

𝑘+1(a, u) ⊆(M𝑖

𝑗)

𝑘(a , u)

Proposition 2.1:

Let G= (V, E) be a connected graph , then M-contractible neighborhood

satisfy

(M𝑖𝑗)

𝑘+1(a, u) ⊆ (M𝑖

𝑗)

𝑘(a , u)⊆ N𝑖

𝑗(𝑎, 𝑢)

Proof

First, since the graph is connected, then the graph is enumerated. Then if

𝑣𝑖 ∈ (M𝑖𝑗)

𝑘+1 and there is edge 𝒆𝒗𝒊 𝒗𝒊+𝟏

∈ 𝐸(𝑉), so if we contract 𝒆𝒗𝒊 𝒗𝒊+𝟏 , then

we eliminate evi vi+1from (Mi

j)

k+1 so |V ((Mi

j)

k+1)| ≤ |V ((Mi

j)

k)| and

|E ((Mij)

k+1)| ≤ |E ((Mi

j)

k)| so (Mi

j)

k+1(a , u) ⊆ (Mi

j)

k(a , u)

Second is obvious.

Proposition 2.2:

Let G= (V, E) be a connected graph then the topological space of

2850 M. Shokry

τk+1(a, u) generating by all M-contractible neighborhood is a sub- topology of

τk(a , u).

(τk+1(a, u) ⊆ τk(a , u) )

Proof

Is obviously from Proposition 2.1

Proposition 2.3:

Let G= (V, E) be a connected graph and τ is topology on G with set of

basis

{{ei}, Nij (a, u)}, ( O 𝑖

𝑗( 𝑎 , 𝑢) ) open set in topology formed on a graph

then

i. ∑|M𝑖𝑗( 𝑎 , 𝑢)| ≤ ∑|N𝑖

𝑗( 𝑎 , 𝑢)|

ii. For any open set contained the deletion edge in topology

∑|CL( O 𝑖 𝑗( 𝑎 , 𝑢) )𝝉𝒌+𝟏

| ≤ ∑|CL( O 𝑖 𝑗( 𝑎 , 𝑢) )𝝉𝒌

|

iii. ∑|int ( O 𝑖 𝑗( 𝑎 , 𝑢) )𝝉𝒌+𝟏

| ≤ ∑|int ( O 𝑖 𝑗( 𝑎 , 𝑢) )𝝉𝒌

|

Proof

Is obvious from Proposition (2.1, 2.2)

3 Topology Induced by Vertices Deletion or Edges Deletion Suppose that G =( V , E) be a graph. If we delete a subset V1 of the set V

and all the edges, which have a vertex in V1 as an end, then the resultant graph is

termed as vertex deleted sub graph of G , so G – e ij ≡ G′ = (V′, E′) ; eij = {ui ,

vj } is a result graph after deletion with vertex where

V′ = {ui : ui ∈ V ; ui ≠ vj } and E′ = 𝐸(𝐺 − 𝑒𝑖𝑗 ) = 𝐸(𝐺) − 𝑒𝑖𝑗 .

Fig (3.1)


The operation of deleting vertex not only removes the vertex v but remove

every edge of which v is end point G – v .We generalized these concepts by

forming new topological properties illustrated the relationship between them by

used a new methods .We generated topology by D- Deleting vertex sets

(D𝑖𝑗)

𝑘(𝑎, 𝑢)as follows.

Definition 3.1

Let G= (V, E) be a graph and 𝐻𝑖𝑗 ⊆ G − v a subgraph generated by all paths

with length j from vertex a and length i from u in G with deletion vertex v, the D-

Deleting vertex neighborhood is defined as

(Dij)

k(a, u)=

{vk, ek : vk ∈ V(Hij), ek ∈ E(Hij), Hij ⊆ G − v , d(vk, u) ≤ i , d (vk, a) ≤ j }

Example3.1

We constructed topological space on comb-graph by used

{∅, (D𝑖𝑗)

𝑘(a, u)} as set of basis. We will begin to apply three operation (D-

Deleting vertex) on it.

Fig (3.2)

As shown in figure we determine the vertex which will be deleted and its

incident edge. Then find the neighborhood and construct the topology.

(D11)1( (a , u) = {{b , e1 },{d , e3}}

(D21)1( (a , u) = {{b , e1 },{d , e5 , f, e3 } , {b , e2 , d, e3 }}

(D12)1( (a , u) = {{b , e1 , d , e2 } , {d , e3}}

(D22)1((a , u) = {{b , e1 , d , e2 } , {d , e5 , f, e3 } , {b , e2 , d, e3 }}

(D13)1( (a , u) = {b , e1 , d , e2 , f ,e5 },{d , e3}}

(D23)1( (a , u) = {{b , e1 , d , e2 , f ,e5 } , { d , e5 , f, e3 } , {b , e2 , d, e3 }}

2852 M. Shokry

Fig (3.3)

The basis β1 = {{e1 } ,{e2 } ,{e3} ,{e5} ,{b} ,{d } ,{b,e1}, {d,e3}, {d ,e5 ,f, e3 } ,

{b,e2 ,d,e3},{b, e1 , d , e2 } ,{b,e1 , d, e2, f ,e5} , {d , b , e2} , {d , f,

e5}}.

(D11)2 (a, u) = {{b , e1 },{d , e3}} , (D2

1)2 (a , u) = {{b , e1 } , {b , e2 , d, e3 }}

(D12)2 (a, u) = {{b , e1 , d , e2 } , {d , e3}}, (D2

2)2 (a , u) = {{b , e1 , d ,e2},{b , e2 , d,

e3 }}

The set of basis

β2 = {{e1 },{e2},{e3},{b},{d },{b,e1},{d,e3},{b,e2 ,d,e3},{b, e1 ,d ,e2} , {d ,b,e2}}.

Fig (3.4)

(D11)3 (a, u) = {{b, e1}}. The set of basis β = {{e1},{e3} , {b,e1} }.

Finally, it's clear from the previous example after applying the method of

deleting vertex on the graph. We will find in the end that if the graph is connected

then we will find similarity of that result from topological space after operations

of M- Contraction edges and D - Deletion vertex.


Proposition 3.1:

Let G= (V, E) be a connected graph. Then D - Deletion of vertex

neighborhood satisfies (D𝑖𝑗)

𝑘+1(a , u) ⊆(D𝑖

𝑗)

𝑘(a , u) ⊆ (N𝑖

𝑗) (a , u)

Proof Is obvious

Proposition 3.2:

Let G= (V, E) be a connected graph. Then topological spaces generated by (𝐃𝒊𝒋)

𝒌

satisfies 𝜏𝑘+1(a, u) is a sub- topology of 𝜏𝑘(a, u).

Proof Is obvious.

Proposition 3.3:

Let G= (V, E) be a connected graph and τ is topology on G with set of basis

{{ei}, Dij (a, u)}, ( O 𝑖

𝑗( 𝑎 , 𝑢) ) open set in topology formed on a graph then

the following satisfies

i. ∑|D𝑖𝑗( 𝑎 , 𝑢)| ≤ ∑|N𝑖

𝑗( 𝑎 , 𝑢)|

ii. Let ( O 𝑖 𝑗( 𝑎 , 𝑢) ) be open sets in topology on a graph then any open set

contained the deletion edge in topology satisfied

(𝑎 ) ∑|CL( O𝑖𝑗(𝑎 , 𝑢) )𝝉𝒌+𝟏

| ≤ ∑|CL( O𝑖𝑗(𝑎 , 𝑢) )𝝉𝒌

|

(𝑏 ) ∑|int ( O𝑖𝑗(𝑎 , 𝑢) )𝝉𝒌+𝟏

| ≤ ∑|int ( O𝑖𝑗(𝑎 , 𝑢) )𝝉𝒌

|

Proof:

i- From proposition 3.1

ii- From O 𝑖 𝑗( 𝑎 , 𝑢) ⊆ O 𝑖

𝑗( 𝑜 , 𝑢) and from proposition 3.2 we obtain the

result obviously.

Suppose that G=(V,E) be a graph. If a subset E1 of the set E and all

incident vertices are deleted from the graph G=(V,E) , then the resultant graph is

termed as edge deleted subgraph G′ = (V′, E′) of G=(V,E) where 𝑉′ = 𝑉(𝐺)

and

𝐸′ = 𝐸(𝐺 − 𝑒𝑖𝑗 ) = 𝐸(𝐺) − 𝑒𝑖𝑗

If G′ a graph resulting from G then a family of all distance neighborhoods

may be compute some topological hereditary properties from G.

2854 M. Shokry

Fig (3.5)

Definition 3.2

Let G= (V, E) be a graph and 𝐻𝑖𝑗 ⊆ G − e a subgraph generated by all paths

with length j from vertex a and length i from u in G with deletion edge e , the D-

Deleting edge neighborhood is defined as

(Lij)

k(a, u)=

{vk, ek : vk ∈ V(Hij), ek ∈ E(Hij), Hij ⊆ G − e , d(vk, u) ≤ i , d (vk, a) ≤ j }

Example3.2

We constructed topological space on comb-graph by using

{∅, (𝐋𝒊𝒋)

𝒌(a, u)} as set of basis. We will begin to apply three operations

(L- Deleting edge) on it.

Fig (3.6)


The operation of deleting edge removes only that edge, the resulting graph

(G – d) or (G-uv).

As shown in figure we determine the edge which be deleted. Then find the

neighborhood and construct the topology.

(L11 )1( (a , u) = {{b , e1 },{d , e3}}

(L21 )1( (a , u) = {{b , e1 },{d , e5 , f, e3 } , {b , e2 , d, e3 }}

(L12 )1( (a , u) = {{b , e1 , d , e2 } , {d , e3}}

(L22 )1((a , u) = {{b , e1 , d , e2 } , {d , e5 , f, e3 } , {b , e2 , d, e3 }}

(L13 )1( (a , u) = {b , e1 , d , e2 , f ,e5 },{d , e3}}

(L23 )1( (a , u) = {{b , e1 , d , e2 , f ,e5 } , { d , e5 , f, e3 } , {b , e2 , d, e3 }}

The set of basis β1 = {{e1 } ,{e2 } ,{e3} ,{e5} ,{b} ,{d } ,{b,e1}, {d,e3}, {d ,e5 ,f, e3

}, {b,e2 ,d,e3},{b, e1 , d , e2 } ,{b,e1 , d, e2, f ,e5} , {d , b , e2} , {d , f, e5}}.

Fig (3.7)

(L11 )2 (a, u) = {{b, e1 }, {d , e3}}

(L21 )2 (a, u) = {{b, e1}, {b, e2, d, e3 }}

(L12 )2 (a, u) = {{b, e1, d, e2}, {d , e3}}

(L22 )2 (a, u) = {{b, e1, d, e2},{b , e2 , d, e3 }}

The set of basis

β2 = {{e1 },{e2},{e3},{b},{d },{b,e1},{d,e3},{b,e2 ,d,e3},{b, e1 ,d ,e2},{d ,b,e2} }.

2856 M. Shokry

Fig (3.8)

(L11 )3 (a, u) = {{b, e1}, {d , e3}}

The set of basis β3 = { {e1 } ,{e3} , {b,e1} ,{d , e3} }.

Finally we will notice from the previous example after applying the

method of deleting edge on the graph. We will find in the end the graph G is

disconnected graph. Since there is no path between the vertices. But also we will

notice that the result topology (V,τ) is a connected space.

Proposition 3.4

Let G= (V, E) be a connected graph, then L - Deletion of edge

neighborhood satisfy (L𝑖𝑗)

𝑘+1(a, u) ⊆(L𝑖

𝑗)

𝑘(a , u) ⊆ (N𝑖

𝑗) (a , u)

Proof Is obvious.

Proposition 3.5

Let G= (V, E) be a connected graph then topological spaces generated by (𝐋𝒊𝒋)

𝒌

satisfies that 𝜏𝑘+1(a , u) is a sub- topology of 𝜏𝑘(a , u).

Proof is obvious.

Proposition 3.6

Let G= (V, E) be a connected graph and τ is topology on G with set of

basis

{{ei}, Lij (a , u)} , ( O 𝑖

𝑗( 𝑎 , 𝑢) ) open set in topology formed on a graph. Then


i-∑|L𝑖𝑗( 𝑎 , 𝑢)| ≤ ∑|N𝑖

𝑗( 𝑎 , 𝑢)|

ii - For any open set contained the deletion edge in topology

∑|CL( O𝑖𝑗(𝑎 , 𝑢) )𝝉𝒌+𝟏

| ≤ ∑|CL( O𝑖𝑗(𝑎 , 𝑢) )𝝉𝒌

|

iii- ∑|int ( O𝑖𝑗(𝑎 , 𝑢) )𝝉𝒌+𝟏

| ≤ ∑|int ( O𝑖𝑗(𝑎 , 𝑢) )𝝉𝒌

|

Proof: Obviously

Conclusion

This research aims to improve comparison between different method of

generated topology based on graph operations. Consequently, we introduce a

modification of some topological concepts by using these new classes. So this

research is considered a starting point of many works in the real life applications.

References

[1] J. Bondy, D. S. Murty, Graph theory with applications, North- Holland, 1992.

[2] R. Diestel, Graph theory II, Springer- Verlag, 2005.

[3] J. Dugundji, Topology, University of Southern California, Los Anageles,

Allyn and Bacon Inc., Boston, Mass, 1966.

[4] S. T.Hu, General Topology, Third Printing – JULY, 1969.

[5] J. R. Munkres, Topology, Prentice- Hall, Inc., Englewood Cliffs, New Jersey,

1975.

[6] R. J. Wilson, Introduction to Graph Theory, Longman Malaysia, 1996.

[7] K. Kutnar, A. Malnic, D. Marusic, S. Milavic. Distance balanced graph:

symmetry conditions, Discrete Mathematics, 306, 1881-1894(2006).

http://dx.doi.org/10.1016/j.disc.2006.03.066

Received: February 2, 2015; Published: April 12, 2015

http://dx.doi.org/10.1016/j.disc.2006.03.066

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Generating Topology on Graphs by Operations on Graphs