Generating Topology on Graphs by. Operations on Graphs

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1 Applied Mathematical Sciences, Vol. 9, 2015, no. 57, HIKARI Ltd, 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

2 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.

3 Generating topology on graphs by operations on graphs 2845 We say that G is distance balanced if v k V(G): d(v k, u) d (v k, v) = v k V(G): d(v k, v) d (v k, 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 B j i (a, u) = {v k V(G): d(v k, a) = j and d (v k, u) = i} The sets B j i (a, u) give rise to a distance partitions of V(G) with respect to i the edge e au E(G) We say that X is strongly distance-balanced if B i 1 (a, u) = B i 1 i (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 N i j (a,u), which contained each vertex and its incident edge which linked to two fixed vertices according to their distance. The topology built by N i j (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 H ij G a subgraph generated by all paths with length j from vertex a and length i from u N i j (a, u) = {v k, e k : v k V(H ij ), e k E(H ij ), H ij G, d(v k, u) i, d (v k, a) j }

4 2846 M. Shokry Example 2.1 a Let G= (V, E) be a comb graph b d e1 e2 e3 u c e4 e5 f Fig (2.1) Firstly, we evaluate the neighborhood of the two fixed vertices v(a) and v(u) : N 1 1 (a, u) = {{b, e1}, {d, e3}} N 2 1 (a, u) = {{b, e1 },{d, e5, f, e3 }, {b, e2, d, e3 }} N 1 2 (a, u) = {{b, e1, c, e4 },{b, e1, d, e2 }, {d, e3}} N 2 2 (a, u) = {{b, e1, c, e4 },{b, e1, d, e2 }, {d, e5, f, e3 }, {b, e2, d, e3 }} N 3 1 (a, u) = {{b, e1}, {d, e3, b, e2, c, e4}} N 1 3 (a, u) = {b, e1, d, e2, f, e5},{d, e3}} N 3 2 (a, u) = {{b, e1, c, e4 },{b, e1, d, e2 },{d, e3, b, e2, c, e4 }} N 2 3 (a, u) = {{b, e1, d, e2, f,e5 }, { d, e5, f, e3 }, {b, e2, d, e3 }} N 3 3 (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 }.

5 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} = } {v e w x w E {e} or yw E {e}} Fig (2.2) Definition 2.3 Let G= (V, E) be a graph and H ij 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 M i j (a, u) = {v k, e k : v k V(H ij ), e k E(H ij ), H ij G/e, d(v k, u) i, d (v k, 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.

6 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 {, (M i j ) 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 (M 1 1 ) 1 ( (a, u) = {{b, e1 },{d, e3}} (M 2 1 ) 1 ( (a, u) = {{b, e1 },{d, e5, f, e3 }, {b, e2, d, e3 }} (M 1 2 ) 1 ( (a, u) = {{b, e1, d, e2 }, {d, e3}} (M 2 2 ) 1 ((a, u) = {{b, e1, d, e2 }, {d, e5, f, e3 }, {b, e2, d, e3 }} (M 1 3 ) 1 ( (a, u) = {b, e1, d, e2, f,e5 },{d, e3}} (M 2 3 ) 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)

7 Generating topology on graphs by operations on graphs 2849 (M 1 1 ) 2 (a, u) = {{b, e1 },{d, e3}}, (M 2 1 ) 2 (a, u) = {{b, e1 }, {b, e2, d, e3 }} (M 1 2 ) 2 (a, u) = {{b, e1, d, e2 }, {d, e3}},(m 2 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 (M 1 1 ) 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 (M i j ) k+1 (a, u) (M i j ) k (a, u) Proposition 2.1: satisfy Proof Let G= (V, E) be a connected graph, then M-contractible neighborhood (M i j ) k+1 (a, u) (M i j ) k (a, u) N i j (a, u) First, since the graph is connected, then the graph is enumerated. Then if v i (M i j ) k+1 and there is edge e vi v i+1 E(V), so if we contract e vi v i+1, then we eliminate e vi v i+1 from (M i j ) k+1 so V ((M i j ) k+1 ) V ((M i j ) k ) and E ((M i j ) k+1 ) E ((M i j ) k ) so (M i j ) k+1 (a, u) (M i j ) k (a, u) Second is obvious. Proposition 2.2: Let G= (V, E) be a connected graph then the topological space of

8 2850 M. Shokry τ k+1 (a, u) generating by all M-contractible neighborhood is a sub- topology of τ k (a, u). Proof Proposition 2.3: (τ k+1 (a, u) τ k (a, u) ) Is obviously from Proposition 2.1 Let G= (V, E) be a connected graph and τ is topology on G with set of basis {{ei}, N j i (a, u)}, ( O j i ( a, u) ) open set in topology formed on a graph then i. M i j ( a, u) N i j ( a, u) ii. For any open set contained the deletion edge in topology CL( O j i ( a, u) ) τk+1 CL( O j i ( a, u) ) τk iii. Proof int ( O j i ( a, u) ) τk+1 int ( O j i ( a, u) ) τk 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 = {u i : u i V ; u i v j } and E = E(G e ij ) = E(G) e ij. Fig (3.1)

9 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 (D i j ) k (a, u)as follows. Definition 3.1 Let G= (V, E) be a graph and H ij 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 (D i j ) k (a, u)= {v k, e k : v k V(H ij ), e k E(H ij ), H ij G v, d(v k, u) i, d (v k, a) j } Example3.1 We constructed topological space on comb-graph by used {, (D i j ) 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. (D 1 1 ) 1 ( (a, u) = {{b, e1 },{d, e3}} (D 2 1 ) 1 ( (a, u) = {{b, e1 },{d, e5, f, e3 }, {b, e2, d, e3 }} (D 1 2 ) 1 ( (a, u) = {{b, e1, d, e2 }, {d, e3}} (D 2 2 ) 1 ((a, u) = {{b, e1, d, e2 }, {d, e5, f, e3 }, {b, e2, d, e3 }} (D 1 3 ) 1 ( (a, u) = {b, e1, d, e2, f,e5 },{d, e3}} (D 2 3 ) 1 ( (a, u) = {{b, e1, d, e2, f,e5 }, { d, e5, f, e3 }, {b, e2, d, e3 }}

10 2852 M. Shokry Fig (3.3) The basis β 1 = {{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}}. (D 1 1 ) 2 (a, u) = {{b, e1 },{d, e3}}, (D 2 1 ) 2 (a, u) = {{b, e1 }, {b, e2, d, e3 }} (D 1 2 ) 2 (a, u) = {{b, e1, d, e2 }, {d, e3}}, (D 2 2 ) 2 (a, u) = {{b, e1, d,e2},{b, e2, d, e3 }} The set of basis β 2 = {{e1 },{e2},{e3},{b},{d },{b,e1},{d,e3},{b,e2,d,e3},{b, e1,d,e2}, {d,b,e2}}. Fig (3.4) (D 1 1 ) 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.

11 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 (D i j ) k+1 (a, u) (D i j ) k (a, u) (N i j ) (a, u) Proof Is obvious Proposition 3.2: Let G= (V, E) be a connected graph. Then topological spaces generated by (D j i ) k satisfies τ k+1 (a, u) is a sub- topology of τ k (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}, D j i (a, u)}, ( O j i ( a, u) ) open set in topology formed on a graph then the following satisfies i. D j i ( a, u) N j i ( a, u) ii. Let ( O j i ( a, u) ) be open sets in topology on a graph then any open set contained the deletion edge in topology satisfied (a ) CL( O j i (a, u) ) τk+1 CL( O j i (a, u) ) τk (b ) int ( O i j (a, u) ) τk+1 int ( O i j (a, u) ) τk Proof: i- From proposition 3.1 ii- From O j i ( a, u) O j i ( 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 and E = E(G e ij ) = E(G) e ij V = V(G) If G a graph resulting from G then a family of all distance neighborhoods may be compute some topological hereditary properties from G.

12 2854 M. Shokry Definition 3.2 Fig (3.5) Let G= (V, E) be a graph and H ij 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 (L i j ) k (a, u)= {v k, e k : v k V(H ij ), e k E(H ij ), H ij G e, d(v k, u) i, d (v k, a) j } Example3.2 We constructed topological space on comb-graph by using {, (L i j ) k (a, u)} as set of basis. We will begin to apply three operations (L- Deleting edge) on it. Fig (3.6)

13 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. (L 1 1 ) 1 ( (a, u) = {{b, e1 },{d, e3}} (L 2 1 ) 1 ( (a, u) = {{b, e1 },{d, e5, f, e3 }, {b, e2, d, e3 }} (L 1 2 ) 1 ( (a, u) = {{b, e1, d, e2 }, {d, e3}} (L 2 2 ) 1 ((a, u) = {{b, e1, d, e2 }, {d, e5, f, e3 }, {b, e2, d, e3 }} (L 1 3 ) 1 ( (a, u) = {b, e1, d, e2, f,e5 },{d, e3}} (L 2 3 ) 1 ( (a, u) = {{b, e1, d, e2, f,e5 }, { d, e5, f, e3 }, {b, e2, d, e3 }} The set of basis β 1 = {{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) (L 1 1 ) 2 (a, u) = {{b, e1 }, {d, e3}} (L 2 1 ) 2 (a, u) = {{b, e1}, {b, e2, d, e3 }} (L 1 2 ) 2 (a, u) = {{b, e1, d, e2}, {d, e3}} (L 2 2 ) 2 (a, u) = {{b, e1, d, e2},{b, e2, d, e3 }} The set of basis β 2 = {{e1 },{e2},{e3},{b},{d },{b,e1},{d,e3},{b,e2,d,e3},{b, e1,d,e2},{d,b,e2} }.

14 2856 M. Shokry Fig (3.8) (L 1 1 ) 3 (a, u) = {{b, e1}, {d, e3}} The set of basis β 3 = { {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 (L i j ) k+1 (a, u) (L i j ) k (a, u) (N i j ) (a, u) Proof Is obvious. Proposition 3.5 Let G= (V, E) be a connected graph then topological spaces generated by (L j i ) k satisfies that τ k+1 (a, u) is a sub- topology of τ k (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}, L j i (a, u)}, ( O j i ( a, u) ) open set in topology formed on a graph. Then

15 Generating topology on graphs by operations on graphs 2857 i- L i j ( a, u) N i 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: 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, [2] R. Diestel, Graph theory II, Springer- Verlag, [3] J. Dugundji, Topology, University of Southern California, Los Anageles, Allyn and Bacon Inc., Boston, Mass, [4] S. T.Hu, General Topology, Third Printing JULY, [5] J. R. Munkres, Topology, Prentice- Hall, Inc., Englewood Cliffs, New Jersey, [6] R. J. Wilson, Introduction to Graph Theory, Longman Malaysia, [7] K. Kutnar, A. Malnic, D. Marusic, S. Milavic. Distance balanced graph: symmetry conditions, Discrete Mathematics, 306, (2006). Received: February 2, 2015; Published: April 12, 2015

Domination Number of Jump Graph

International Mathematical Forum, Vol. 8, 013, no. 16, 753-758 HIKARI Ltd, www.m-hikari.com Domination Number of Jump Graph Y. B. Maralabhavi Department of Mathematics Bangalore University Bangalore-560001,