Open access peer-reviewed chapter

Folding on the Chaotic Graph Operations and Their Fundamental Group

Written By

Mohammed Abu Saleem

Reviewed: July 11th, 2019 Published: October 25th, 2019

DOI: 10.5772/intechopen.88553

From the Edited Volume

Functional Calculus

Edited by Kamal Shah and Baver Okutmuştur

Chapter metrics overview

774 Chapter Downloads

View Full Metrics


Our aim in the present chapter is to introduce a new type of operations on the chaotic graph, namely, chaotic connected edge graphs under the identification topology. The concept of chaotic foldings on the chaotic edge graph will be discussed from the viewpoint of algebra and geometry. The relation between the chaotic homeomorphisms and chaotic foldings on the chaotic connected edge graphs and their fundamental group is deduced. The fundamental group of the limit chaotic chain of foldings on chaotic. Many types of chaotic foldings are achieved. Theorems governing these relations are achieved. We also discuss some applications in chemistry and biology.


  • chaotic graph
  • edge graph
  • chaotic folding
  • limit folding fundamental group
  • 2010 Mathematics Subject Classification: 51H20
  • 57N10
  • 57M05
  • 14F35
  • 20F34

1. Introduction and definitions

During the past few decades, examinations of social, biological, and communication networks have taken on enhanced attention throughout these examinations; graphical representations of those networks and systems have been evident to be terribly helpful. Such representations are accustomed to confirm or demonstrate the interconnections or relationships between parts of those networks [1, 2].

A graph is an ordered G = (V(G), E(G)) where V(G) ≠ φ, E(G) is a set disjoint from V(G), elements of V(G) are called the vertices of G, and elements of E(G) are called the edges. The foundation stone of graph theory was laid by Euler in 1736 by solving a puzzle called Königsberg seven-bridge problem as in Figure 1 [1, 3].

Figure 1.

Königsberg seven-bridge problem.

There are many graphs with which one can construct a new graph from a given graph or set of graphs, such as the Cartesian product and the line graph. A graph G is a finite non-empty set V of objects called vertices (the singular is vertex) together with a set E of two-element subsets of V called edges. The number of vertices in a graph G is the order of G, and the number of edges is the size of G. To indicate that a graph G has vertex set V and edge set E, we sometimes write G = (V, E). To emphasize that V is the vertex set of a graph G, we often write V as V(G). For the same reason, we also write E as E(G). A graph H is said to be a subgraph of a graph G if V(H) ⊆ V(G) and E(H) ⊆ E(G). The complete graph with n-vertices will be denoted by Kn. A null graph is a graph containing no edges; the null graph with n-vertices is denoted by Nn. A cycle graph is a graph consisting of a single cycle, the cycle graph with n-vertices is denoted by Cn. The path graph is a graph consisting of a single path; the path graph with n-vertices is denoted by Pn [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Let G and H be two graphs. A function φ:VGVH is a homomorphism from G to H if it preserves edges, that is, if for every edge eEG,feEH [12, 13]. A core is a graph which does not retract to a proper subgraph. Any graph is homomorphically equivalent to a unique core [7].

The folding is a continuous function f:GH such that for each vVG,fvVH, and for each eEG,feEH [14]. Let X be a space, and let I be the unit interval [0,1] in R, a homotopy of paths in X is a family gt:IX,0t1such that (i) the endpoints gt0=x0 and gt1=x1 are independent of t and (ii) the associated map G:I×IX defined by G(s,t) = gt(s) is continuous [15]. Given spaces X and Y with chosen points x0X, and y0Y, the wedge sum X∨Y is the quotient of the disjoint union X∪Y obtained identifying x0 and y0 to a single point [15]. Two spaces X and Y are of the “same homotopy type” if there exist continuous maps f:XY and g:YX such that gfIX:XX and fgIY:YY [16]. The fundamental group briefly consists of equivalence classes of homotopic closed paths with the law of composition following one path to another. However, the set of homotopy classes of loops based at the point x0 with the product operation fg=f·g is called the fundamental group and denoted by π1Xx0 [4, 17, 18, 19, 20, 21, 22, 23, 24]. Over many years, chaos has been shown to be an interesting and even common phenomenon in nature. Chaos has been shown to exist in a wide variety of settings: in fluid dynamics such as Raleigh-Bernard convection, in chemistry such as the Belousov-Zhabotinsky reaction, in nonlinear optics in certain lasers, in celestial mechanics, in electronics in the flutter of an overdriven airplane wing, some models of population dynamics, and likely in meteorology, physiological oscillations such as certain heart rhythms, as well as brain patterns [17, 24, 25, 26, 27, 28, 29, 30]. AI algorithms related to adjacency matrices on the operations of the graph are discussed in [31, 32].


2. The main results

First, we will introduce the following:

Definition 1. The chaotic edgee¯is a geometric edgee1that carries many other edgese2e3, each one of them homotopic to the original one as in Figure 2. Also the chaotic vertices ofe¯arev¯=v1v2andu¯=u1u2. For chaotic edgee¯, we have two cases:

Figure 2.

Chaotic edge.

Case 1 (1) e1,e2,e3,are of the same physical properties.

Case 2 (2) e1,e2,e3,represent different physical properties; for example,e1represents density,e2represents hardness,e3represents magnetic fields, and so on.

Definition 2. A chaotic graphG¯is a collection of finite non-empty setV¯of objects called chaotic vertices together with a setE¯of two-element subsets ofV¯called chaotic edges. The number of chaotic edges is the size ofG¯.

Definition 3. Given chaotic connected graphsG¯1andG¯2with given edgese¯1G¯1ande¯2G¯2, then the chaotic connected edge graphG¯1G¯2is the quotient of disjoint unionG¯1G¯2acquired by identifying two chaotic edgese¯1ande¯2to a single chaotic edge (up to chaotic isomorphism) as in Figure 3.

Figure 3.

Chaotic connected edge.

Definition 4. A chaotic graphH¯is called a chaotic subgraph of a chaotic graphG¯ifV¯H¯V¯G¯andE¯H¯E¯G¯.

Definition 5. LetG¯andH¯be two chaotic graphs. A functionφ¯:V¯G¯V¯H¯is chaotic homomorphism fromG¯toH¯if it preserves chaotic edges, that is, if for any chaotic edgeu¯v¯ofG¯,φ¯u¯φ¯v¯is a chaotic edge ofH¯.

Definition 6. A chaotic folding of a graphG¯is a chaotic subgraphH¯ofG¯such that there exists a chaotic homomorphismf¯:G¯H¯,called chaotic folding withf¯x¯=x¯for every chaotic vertexx¯ofH¯.

Definition 7. A chaotic core is a chaotic graph which does not chaotic retract to chaotic proper subgraph.

Theorem 1. LetG¯1andG¯2be two chaotic connected graphs. Thenπ¯1G¯1G¯2=π¯1G¯1π¯1G¯2.

Proof. Let G¯1 and G¯2 be two chaotic connected graphs. Since G¯1G¯2 and G¯1G¯2 are of same chaotic homotopy type, it follows that π¯1G¯1G¯2π¯1G¯1π¯1G¯2. Hence, π¯1G¯1G¯2=π¯1G¯1π¯1G¯2.

Theorem 2. The chaotic graphsG¯1andG¯2are chaotic subgraphs ofG¯1G¯2. Also, for any chaotic treeG¯1andG¯2,G¯1G¯2is also chaotic tree andπ¯1G¯1G¯2=0¯.

Proof. The proof of this theorem is clear.

Theorem 3. IfG¯1,G¯2,,G¯nare connected graphs, andf¯1f¯2f¯nis a sequence of chaotic topological foldings ofi=1nG¯iinto itself, then there is an induced sequencef¯¯1f¯¯2f¯¯nof non-trivial chaotic topological foldingf¯¯j:i=1nπ¯1G¯iii=1nπ¯1G¯ii,j=1,2,,nsuch thatf¯¯ji=1nπ¯1G¯iireduces the rank ofi=1nπ¯1G¯ii.

Proof. Consider the following sequence of topological foldings f¯1f¯2f¯n, where f¯1:i=1nG¯ii=1nG¯i, is a topological folding from i=1nG¯i into itself such that f¯1i=1nG¯i=G¯1G¯2f¯1G¯sG¯nfors=1,2,n.

Since sizef¯1G¯ssizeG¯s and f¯¯1π¯1G¯i=π¯1f¯1G¯i, it follows that rankf¯¯1π¯1G¯s=rankπ¯1f¯1G¯srankπ¯1G¯s, and so f¯¯1 reduces the rank of i=1nπ¯1G¯ii. Also, if f¯2i=1nG¯i=G¯1G¯2f¯2G¯sf¯2G¯kG¯nfork=1,2,nands<k and sizef¯2G¯ssizeG¯sand sizef¯2G¯ksizeG¯k, we haverankf¯¯2π¯1G¯s=rankπ¯1f¯2G¯srankπ¯1G¯s, rankf¯¯2π¯1G¯k=rankπ¯1f¯2G¯krankπ¯1G¯k; thus f¯¯2 reduces the rank of i=1nπ¯1G¯ii. Moreover, by continuing with this procedure if f¯ni=1nG¯i=i=1nf¯nG¯i, then f¯¯ni=1nπ¯1G¯ii=π¯1f¯ni=1nG¯i=π¯1i=1nf¯nG¯ii=1nπ¯1f¯nG¯ii. Hence, f¯¯n reduces the rank of i=1nπ¯1G¯ii.

Theorem 4. LetG¯1andG¯2be two chaotic graphs; then there is a chaotic homomorphismf¯:G¯1G¯2which inducesf¯¯:π¯1G¯1π¯1G¯2ifπ¯1G¯2is a chaotic folding ofπ¯1G¯1G¯2.

Proof. Let f¯:G¯1G¯2 be a chaotic homomorphism. Since G¯2 is chaotic subgraph of G¯1G¯2, then there exists a chaotic homomorphism f¯:G¯1G¯2G¯2 with f¯v¯=v¯ for any chaotic vertex v¯ of G¯2 which induces f¯¯:π¯1G¯1π¯1G¯2. What follows from G¯2 is a chaotic folding of G¯1G¯2 in that π¯1G¯2 is a chaotic folding of π¯1G¯1G¯2. Conversely, assume that G¯2 is a chaotic folding of G¯1G¯2; thus f¯:G¯1G¯2G¯2 is a chaotic homomorphism with f¯v¯=v¯ for any chaotic vertex v¯ of G¯2, and so there is a chaotic homomorphism f¯:G¯1G¯2 which induce f¯¯:π¯1G¯1π¯1G¯2.

Theorem 5. For any chaotic path graphsP¯n,P¯m,n,m2,there is a sequence of topological foldings with variation curvaturef¯i:i=12konP¯nP¯mwhich induce a sequence of topological foldingsf¯¯i:i=12ksuch thatf¯¯kf¯¯k1f¯¯1π1P¯nP¯m=Z¯andlimkf¯¯kf¯¯k1f¯¯1π1P¯nP¯m=0¯.

Proof. Consider the following sequence of chaotic topological foldings with variation curvature, f¯1:P¯nP¯mP¯nP¯m1, where P¯nP¯m1 is a chaotic subgraph with decreasing inner curvature between every two adjacent chaotic edges in P¯nP¯m and f¯2:f¯1P¯nP¯mf¯1P¯nP¯m1 where f¯2f¯1P¯nP¯m1 is a chaotic subgraph with decreasing inner curvature between every two adjacent chaotic edges in f¯1P¯nP¯m1, and so on, such that f¯kf¯k1f¯k2f¯1P¯nP¯m=C¯n+m2 and limkf¯kf¯k1f¯k2f¯1P¯nP¯m=N¯1, thus f¯¯kf¯¯k1f¯¯k2f¯¯1π1P¯nP¯m=π¯11C¯n+m2=Z¯. Also, limkf¯¯kf¯¯k1f¯¯k2f¯¯1π¯1P¯nP¯m=π¯1N¯1=0¯.

Theorem 6. For every two chaotic connected graphsG¯1andG¯2, the fundamental group of the limit of chaotic topological folding ofG¯1G¯2=0¯.

Proof. Let G¯1 and G¯2 be two chaotic connected graphs; then we have two cases:

Case (1): If f¯1:G¯1G¯2G¯1G¯2 is a chaotic topological folding such that f¯1G¯1G¯2 consists of chaotic cycles, so we can define a sequence of chaotic topological folding f¯2:f¯1G¯1G¯2f¯1G¯1G¯2 where f¯2f¯1G¯1G¯2 is a chaotic tree with n¯chaotic edges, f¯3:f¯2f¯1G¯1G¯2f¯2f¯1G¯1G¯2, such that f¯3f¯2f¯1G¯1G¯2 is a chaotic tree with k¯<n¯ chaotic edges, chaotic edges by continuing this process we get f¯k:f¯k1f¯k2f¯1G¯1G¯2f¯k1f¯k2f¯1G¯1G¯2 such that limkf¯kf¯k1f¯k2f¯1G¯1G¯2 is a chaotic edge, and so π¯1limkf¯kf¯k1f¯k2f¯1G¯1G¯2=0¯.

Case (2): If g¯1:G¯1G¯2G¯1G¯2 is a chaotic topological folding such that.

g¯1G¯1G¯2 has no chaotic cycles, then clearly limkg¯kg¯k1g¯k2g¯1G¯1G¯2 is a chaotic edge and π¯1limkg¯kg¯k1g¯k2g¯1G¯1G¯2=0¯.

Theorem 7. IfG¯1andG¯2are chaotic connected and not chaotic cores graphs, thenπ¯1limnf¯nG¯1G¯2=π¯1limnf¯nG¯1π¯1limnf¯nG¯2.

Proof. If G¯1 and G¯2 are chaotic connected and not chaotic cores graphs, then we get the following chaotic induced graphs limnf¯nG¯1G¯2,limnf¯nG¯1,limnf¯nG¯2, and each of them are isomorphic to k¯2. Since k¯2k¯2k¯2 it follows that limnf¯nG¯1G¯2=limnf¯nG¯1limnf¯nG¯2 and π¯1limnf¯nG¯1G¯2 = π¯1limnf¯nG¯1π¯1limnf¯nG¯2.


3. Some applications

  1. A polymer is composed of many repeating units called monomers. Starch, cellulose, and proteins are natural polymers. Nylon and polyethylene are synthetic polymers. Polymerization is the process of joining monomers. Polymers may be formed by addition polymerization; furthermore, one essential advance likewise polymerization is mix as in Figure 4, which happens when the polymer’s development is halted by free electrons from two developing chains that join and frame a solitary chain. The accompanying chart portrays mix, with the image (R) speaking to whatever remains of the chain.

  2. Chemical nature of enzymes, all known catalysts are proteins. They are high atomic weight mixes made up primarily of chains of amino acids connected together by peptide bonds as in Figure 5.

  3. There are two types of the subunit structure of ribosomes as in Figure 6 which is represented by the different connected types of protein subunit and rRNA to form a new type of ribosomes.

Figure 4.


Figure 5.

Typical amino acids.

Figure 6.

Prokaryotic ribosome components.


4. Conclusion

In this chapter, the fundamental group of the limit chaotic foldings on chaotic connected edge graphs is deduced. Also, we can deduce some algorithms from a new operation of a graph by using the adjacency matrices.


  1. 1. Bailey S. Dot Product Graphs and Their Applications to Ecology. Logan, Utah: Utah State University; 2013
  2. 2. Pirzada S, Dharwadker A. Applications of graph theory. The Journal of the Korean Society for Industrial and Applied Mathematics. 2007;11(4):19-38
  3. 3. Bondy JA, Murty USR. Graph Theory. Springer; 2008
  4. 4. Abu-Saleem M. Folding on the wedge sum of graphs and their fundamental group. Applied Sciences. 2010;12:14-19
  5. 5. Abu-Saleem M. Retractions and homomorphisms on some operations of graphs. Journal of Mathematics. 2018:1-4
  6. 6. Biggs NL, Lloyd EK, Wilson RJ. Graph Theory. Oxford: Oxford University Press; 1976. pp. 1736-1936
  7. 7. Chartrand G, Zhang P. Chromatic Graph Theory. First Edition. Taylor & Francis Group; 2009
  8. 8. Giblin J. Graphs, Surfaces, and Homology: An Introduction to Algebraic Topology. Second Edition. London: Chapman and Hall Ltd.; 1977
  9. 9. Rahman MS. Basic Graph Theory. Springer International Publishing AG; 2017
  10. 10. White AT. Graphs, Groups and Surfaces. Amsterdam: North-Holland Publishing Company; 1973
  11. 11. Wilson RJ, Watkins JJ. Graphs, An Introductory Approach, A First Course in Discrete Mathematics. Canada John Wiley & sons, Inc.; 1990
  12. 12. Hammack R, Imrich W, Klavžar S. Handbook of Product Graphs. Boca Raton: Taylor & Francis Group; 2011
  13. 13. Hell P, Nesetfil J. Graphs and Homomorphisms. New York: Oxford University Press; 2004
  14. 14. El-kholy E, El-Esawy A. Graph folding of some special graphs. Journal of Mathematics and Statistics. 2005;1(1):66-70
  15. 15. Massey WS. Algebraic Topology, an Introduction. New York: Harcourt Brace and World; 1967
  16. 16. Jones D. Chaos Math Online. Available from: [Accessed: 11 November 2001]
  17. 17. Abu-Saleem M. Folding on the chaotic Cartesian product of manifolds and their fundamental group. Tamkang Journal of Mathematics. 2008;39:353-361
  18. 18. Abu-Saleem M. Dynamical manifold and their fundamental group. Advanced Studies in Contemporary Mathematics. 2010;20(1):125-131
  19. 19. Abu-Saleem M. Conditional fractional folding of a manifold and their fundamental group. Advanced Studies in Contemporary Mathematics. 2010;20(2):271-277
  20. 20. Abu-Saleem M. Folding on manifolds and their fundamental group. Journal of Taibah University for Science. United Kingdom; 2018:1-4
  21. 21. El-Ghoul M, El-Ahmady AE, Abu-Saleem M. Folding on the Cartesian product of manifolds and their fundamental group. Applied Sciences. Romania; 2007;9:86-91
  22. 22. El-Ghoul M, El-Ahmady AE, Rafat H, Abu-Saleem M. The fundamental group of the connected sum of manifolds and their foldings. International Journal of Pure and Applied Mathematics. 005;22(2):557-567
  23. 23. Frigerio R. Hyperbolic manifold with geodesic boundary which are determine by their fundamental group. Topology and its Applications. 2004;45:69-81
  24. 24. Neto O, Silva PC. The fundamental group of an algebraic link. Comptes Rendus de l’Académie des Sciences, Paris, Série I. 2005;340:141-146
  25. 25. Abu-Saleem M. On dynamical chaotic de sitter spaces and their deformation retracts. Proceedings of the Jangjeon Mathematical Society. 2011;14:231-238
  26. 26. Abu-Saleem M. Dynamical chaotic homotopy group and its applications. International Journal of Nonlinear Science. 2011;11(12):206-212
  27. 27. Alligood KT, Sauer TD, Yorke JA. Chaos: An Introduction to Dynamical Systems. New York, Heidelberg, Berlin: Springer-Verlag; 1990
  28. 28. Devaney RL. An Introduction to Chaotic Dynamical Systems. 2nd ed. New York: Addison-Wesley; 1989
  29. 29. Peitgen H-O, Jürgen H, Saupe D. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag; 1992
  30. 30. Rafat H. Conditional foldings of chaotic manifolds and their retractions [thesis]. Tanta, Egypt: University of Tanta; 2004
  31. 31. Kutschenreiter-Praszkiewicz I. Graph-based decision making in industry. In: Sirmacek B, editor. Graph Theory—Advanced Algorithms and Applications. IntechOpen; 2017. DOI: 10.5772/intechopen.72145. Available from:
  32. 32. Mayer DC. Modeling rooted in-trees by finite p-groups. In: Sirmacek B, editor. Graph Theory—Advanced Algorithms and Applications. United Kingdom: IntechOpen; 2017. DOI: 10.5772/intechopen.68703. Available from:

Written By

Mohammed Abu Saleem

Reviewed: July 11th, 2019 Published: October 25th, 2019