Retractions and Homomorphisms on Some Operations of Graphs
The aim of the present article is to introduce and study a new type of operations on graph, namely, edge graph. The relation between the homomorphisms and retractions on edge graphs is deduced. The limit retractions on the edge graphs are presented. Retractions on a finite number of edge graphs are obtained.
1. Introduction and Preliminaries
Graph theory is rapidly moving into the mainstream of mathematics. The prospects of further development in algebraic graph theory and important link with computational theory indicate the possibility of the subject quickly emerging at the forefront of mathematics. Its scientific and engineering applications, especially to communication science, computer technology, and system theory, have already been accorded a place of pride in applied mathematics. Graphs serve as mathematical models to analyze successfully many concrete real-world problems. A certain problem in physics, chemistry, genetics, psychology, sociology, and linguistics can be formulated as problems in graph theory. Also, many branches of mathematics such as game theory, group theory, matrix theory, probability, and topology have interactions with graph theory. Some puzzles and various problems of a practical nature have been instrumental in the development of various topics in graph theory. The theory of acyclic graphs was developed for solving problems of electrical networks and the study of trees was developed for enumerating isomers of organic compounds. This paper describes the operation of a graph from the viewpoint of an identification [1–10].
A graph is an ordered , where , is a set disjoint from , elements of are called the vertices of , and elements of are called the edges. A graph is connected if, for every partition of its vertex set into two nonempty sets and , there is an edge with one end in and one end in ; otherwise, the graph is disconnected. A graph is said to be a subgraph of a graph if and . A graph in which each pair of distinct vertices is adjacent is called a complete graph. A complete graph with n vertices is denoted by . The chromatic number of a graph is the minimum number of colors required for proper vertex coloring of . A coloring of a graph is a vertex coloring of that uses at most colors. A graph is said to be -colorable if admits a proper vertex coloring using at most colors . Let and be two graphs. A function is a homomorphism from to if it preserves edges, that is, if for any edge of , is an edge of . A retract of a graph is a subgraph of such that there exists a homomorphism , called retraction with for any vertex of . A core is a graph which does not retract to a proper subgraph .
2. The Main Results
Aiming at our study, we will introduce the following.
Definition 1. Let and be two connected graphs, where is an edge of , is an edge of , and ; then we define the edge graph by gluing together the two edges and .
Theorem 2. Let and be two connected graphs. Then
Proof. Let . At that point, there exists an -coloring of Since assigns different colors to every two adjacent vertices of and , and are -colorable and so and Also, using symmetry . Beginning with an ideal coloring of , we can incorporate an ideal coloring of by exchanging a pair of color names to make the coloring agree at two vertices of common edge graphs. This produces a proper coloring of .
Theorem 3. The graphs and are subgraphs of . Also, for any tree and , the graph is also a tree.
Proof. The proof of this theorem is clear.
Theorem 4. Suppose that are connected graphs; then there is a sequence of nontrivial retractions , where are glued along the same edge such that is a proper subgraph of
Proof. Let be a retraction from into itself and for . Since is a proper subgraph of , it follows that is a proper subgraph of . Also, if for and are subgraphs of , respectively, then is a proper subgraph of . Moreover, by continuing this process if , then is a proper subgraph of .
Theorem 5. Let and be two graphs; then there is a homomorphism iff is a retract of .
Proof. Let be a homomorphism. Since is subgraph of , then there exists a homomorphism with , for any vertex of and so is a retract of . Conversely, assume that is a retract of ; thus is a homomorphism with for any vertex of , and so there is a homomorphism .
Theorem 6. Let and be connected graphs; then is a retract of graph or , iff retract of .
Proof. Suppose and are connected graphs and is a retract of graph or . Then there is a homomorphism such that , or a homomorphism such that , for any vertex of Since is a core and it is subgraph of or , it follows that is subgraph of and so there is a homomorphism such that , for any vertex of . Conversely, suppose is a retract ; then is subgraph of and and so is a retract of graphs or .
Theorem 7. Let be any tree of size ; then there is a sequence of nontrivial retractions such that
Proof. Consider the following sequence of retractions:
is nontrivial retraction, where is subgraph of and ,
, where is subgraph of and ,
, where is subgraph of , and is a tree of size . Therefore, .
Theorem 8. Suppose that and are connected graphs; then
Proof. If and are connected graphs, then we get the following induced subgraphs and each of them is isomorphic to . Since, , it follows that
3. Some Applications in Chemistry and Biology
(i) 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 and one basic step in addition polymerization is combination as in Figure 1, which occurs when the polymer’s growth is stopped by free electrons from two growing chains that join and form a single chain. The following diagram depicts combination, with the symbol (R) representing the rest of the chain. This is a representation type of connected two graphs into an edge graph.
(ii) Peptide bonds constitute the representation of an edge graph by linking two amino acids as in Figure 2, which is a representation graph of connected two typical amino acids into an edge graph.
In Figure 3, peptide pond and formation hydrolysis: Formation (top to bottom) and hydrolysis from bottom to top of a peptide bonds require conceptually loss and addition, respectively, of a molecule of water. The actual chemical synthesis and hydrolysis of peptide bonds in the cell are enzymatically controlled processes that in the synthesis nearly always occur on the ribosome and are directed by an mRNA template. The end of a polypeptide with the free of amino group is known as the amino terminus (N terminus) and with the free carboxyl group is the carboxyl terminus (C terminus). This is a representation of connected two graphs into an edge graph.
No data were used to support this study.
Conflicts of Interest
The author declares that they have no conflicts of interest.
A. T. White, Graphs, Groups and Surfaces, North-Holland Publishing Co., Amsterdam, Holland, 1973.View at: MathSciNet
R. J. Wilson and J. J. Watkins, Graphs, An Introductory Approuch, A First Course in Discrete Mahematics, Canada John Wiley and sons, Inc., 1990.View at: MathSciNet
G. Chartrand and P. Zhang, Chromatic Graph Theory, Taylor and Francis group, 2009.View at: MathSciNet