The Graph of Simplex Vertices

Transcription

1 wwwccsenetorg/jmr Journal of Mathematics Research Vol 4, No ; February The Graph of Simplex Vertices M El-Ghoul Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt Tel: m elghoul@hotmailcom N El-Obaidi Mathematics Department, Faculty of Science, Girl s College, Cairo, Egypt Tel: najat alobaidi@yahoocom Received: June 7, Accepted: July 5, Published: February, doi:559/jmrv4np5 URL: Abstract In this paper we will introduce a new types of graph The representation of the new graph by adjacent and incidence matrices will be obtaind Some geometric transformations on the new graphs are described Keywords: Graphs, Simplex Mathematics subject classification: 5H, 57N Definitions and Background () Abstract graph: An abstract graph G is a diagram consisting of a finite non empty set of the elements, called vertices denoted by V(G) together with a set of unordered pairs of these elements, called edge denoted by E(G) The set of vertices of the graph G is called the vertex-set of G and the list of edges is called the edge -list of G (Giblin, 977; Gibbson, 995) () Simplex: Given any set V = {v, v,, v n } of n + points in R n, such that the differences v v, v v,, v n v are linearly independent, the n-simplex with vertices V is the convex hull of V, ie the set of all points of the form t v + t v + + t n v n, where n i= t i = and t i for all i (Hatcher, ) () Adjacency and Incidence: let v and w be vertices of a graph if v and w are joined by an edge e Then v and w are said to be adjacent, moreover, v and w are said to be incident with e, and e is said to be incident with v and w (Wilson, 97) (4) The adjacency matrix: let G be a graph without loops, with n-vertices labeled,,,, n The adjacency matrix A(G) is the nxn matrix in which the entry in row i and column j is the number of edges joining the vertices i and j (Wilson, 97) (5) The incidence matrix: let G be a graph without loops, with n-vertices labeled,,,, n and m edges labeled,,,, m The incidence matrix I(G) is the nxn matrix in which the entry in row i and column j is if vertex i is incident with edge j and otherwise (Wilson & Watkins, 99; Gross & Tucker, 987) (6) Folding and unfolding of graph: (a) Let f : G G be a map between any two graphs G, G and (not necessary to be simple) such that if (u, v) G, ( f (u), f (v)) G Then f is called a topological folding of G to provided that d( f (u), f (v)) d(u, v) (Giblin, 977) (b) Let g : G G be a map between any two graphs G, G and (not necessary to be simple) such that if (u, v) G, (g(u), g(v)) G Then g is called a topological unfolding of G to provided that d(g(u), g(v)) d(u, v)(el-ghoul, 7) Main Result Now we will define and discuss the graph of simplex vertices and some transformations on this new graph, the incident and adjacent matrices which represent these new graphs will be discussed Definition The graph of simplex vertices is a pair (V(G), E(G)) where V(G) = {{V }, {V },, {V i }} is a finite non-empty set of vertices in R n, i =,,,, n and each vertex consistes of k-simplex graph, ie {V i } = {{v i, e i }, {v i, e i },, {v ik, v ik }}and E(G) is Published by Canadian Center of Science and Education 5

2 wwwccsenetorg/jmr Journal of Mathematics Research Vol 4, No ; February a set of unordered pairs of distinct elements of V(G) The graph of -simplex vertices It is represented as a simple graph, see Fig() It s adjacent and incidence are: Where ( ) refer to -simplex graph The graph of -simplex vertices It has two types of dimensions of edges Type() The edge of -dimension see Fig(), Fig(), Fig(4) V(G) = {V = {v, e, v }, V = {v, e, v }}, E(G) = {e } Fig() The adjacent and incidence are: Where refer to the dimension and form of the edge on the graph Fig() Fig(4) Type() The edge of -dimension see Fig(5), Fig(6) Fig(5) The adjacent and incidence are: Fig(6) 4 The graph of -simplex vertices In this case the edge will be in three types of dimensions Type() The edge of -dimension see Fig(7), Fig(8), Fig(9), Fig() Fig(7) The adjacent and incidence will be in the form: 6 ISSN E-ISSN

3 wwwccsenetorg/jmr Journal of Mathematics Research Vol 4, No ; February Fig(8) where Fig(9) Fig() Where ( ) refer to the edge which connecte between the area of each simplex Type() The edge of -dimension see Fig(), Fig(), Fig(), Fig(4) Fig() It s adjacent and incidence are: Fig() Fig() Fig(4) Type() The edge of -dimension see Fig(5), Fig(6) Fig(5) The adjacent and incidence will be in the form: Fig(6) 5 The graph of simplex vertices in higher dimension The graph of n-simplex vertices has types of edges of higher dimensions We can represent these types of edges by matrices as the following: Published by Canadian Center of Science and Education 7

4 wwwccsenetorg/jmr Journal of Mathematics Research Vol 4, No ; February The st n type: n n+ The ed n type: n n The ed n type: n n The (n + ) type: n n n n n n n n n n Lemma Any graph G of n-simplex vertices can be connected by edges of (n +, n, n,, ) dimension n n+ n Lemma The edge which connect between two vertices of simplex graph has dimension equal to or less than one the largest number of -faces that belong to any of the simplex graph Folding of Geometric Graph of Simplex Vertices We can make many types of foldings Type() Folding of External Vertices and External Edges Fig(7) such that I(G) = Where the upper suffix refers to the existence of loops Fig(8) where Fig(9) where I(G) = I(G) = F F F F F F F F F F F F () 8 ISSN E-ISSN

5 wwwccsenetorg/jmr Journal of Mathematics Research Vol 4, No ; February Where () refer to the loop of dimension Fig() I(G) = F F F F Fig() Fig() Fig() I(G) = I(G) = I(G) = F F F F F F F F F F F () F () Some loopes will take different shape see Fig(4) The adjacent and incidence will be in the form: I(G) = F F F () F Type() Folding of internal edges and internal vertices see Fig(5) Where F F F Published by Canadian Center of Science and Education 9

6 wwwccsenetorg/jmr Journal of Mathematics Research Vol 4, No ; February I(G) = F F F Where( ) refer to number of internal edges Fig(6) I(G) = F () () () F F () () F F F () () Fig(7) I(G) = F F F F F F F 4 F 4 Fig(8) I(G) = F ()() ()() F F F F 4 ()() () ()() ()() ()() F F 4 F Type() In the next graph we will fold the length of the edges on each other see Fig(9), Fig() Fig(9) I(G) = F F F F F F Fig() I(G) = F F F F F F ISSN E-ISSN

7 wwwccsenetorg/jmr Journal of Mathematics Research Vol 4, No ; February Type(4) The folding which reduce the volume of the graph, the incidence and adjacent matrices are of the same type see Fig() I(G) = F F F F Lim F n n Lim F n n Theorem The end of the limit of foldings of simplex graph of dimension n is the -simplex graph Proof: Let G is a simplex graph of dimension n, f is a folding f : G G such that f (E ) = E then f (G) = G, dim E = dim E Let f (G ) = G, f (E ) = E, dim E = dim E,, f n (G n ) = G n, f n (E n ) = E n+, lim f n (G n ) = H, H of (n )dimension And g : H H such that g (E ) = E then g (H) = H, dim E = dim E Let g (H ) = H, g (E ) = E, dim E = dim E,, g n (H n ) = H n, g n (E n ) = E n+, lim g n(h n ) = L, L of (n ) dimension And lim lim f n(g n ) = n n -simplex graph see Fig() References El-Ghoul, M Unfolding of graph and uncertain graph The Australian Mathematics Journal, Sandy Bay 76, Tasmania, Australia Gibbson, A (995) Algorithmic graph theory Cambridge University Press, Cambridge, UK Giblin, P J (977) Graphs, surfaces and homology: an introduction to algebraic topology Chapman and Hall, Ltd, London Gross, J L, & Tucker, T W (987) Topological graph theory Jon Wiley & Sons, Inc, Canada Hatcher, A () Algebraic topology Cambridge University Press Wilson, R J (97) Introduction to graph theory Oliver and Boyd, Edinburgh Wilson, R J, & Watkins, J J (99) Graphs: an introductory approach a first course in discrete mathematics Jon Wiley & Sons, Inc, Canada n Figure -4 Published by Canadian Center of Science and Education

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9 wwwccsenetorg/jmr Journal of Mathematics Research Vol 4, No ; February Figure 9- Published by Canadian Center of Science and Education

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11 wwwccsenetorg/jmr Journal of Mathematics Research Vol 4, No ; February Figure 8- Published by Canadian Center of Science and Education 5