Post on 14-Mar-2022
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
Generating topology on graphs by operations on graphs 2847
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)
Generating topology on graphs by operations on graphs 2849
(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)
Generating topology on graphs by operations on graphs 2851
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.
Generating topology on graphs by operations on graphs 2853
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)
Generating topology on graphs by operations on graphs 2855
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
Generating topology on graphs by operations on graphs 2857
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
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Allyn and Bacon Inc., Boston, Mass, 1966.
[4] S. T.Hu, General Topology, Third Printing – JULY, 1969.
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http://dx.doi.org/10.1016/j.disc.2006.03.066
Received: February 2, 2015; Published: April 12, 2015