Retractions and Homomorphisms on Some Operations of Graphs

Graph theory is rapidly moving into the mainstream ofmathematics. 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 orderedG = (V(G), E(G)), whereV(G) ̸ = φ, E(G) is a set disjoint from V(G), elements of V(G) are called the vertices ofG, and elements of E(G) are called the edges. A graph is connected if, for every partition of its vertex set into two nonempty sets X and Y, there is an edge with one end in X and one end in Y; otherwise, the graph is disconnected. A graphH is said to be a subgraph of a graphG ifV(H) ⊆ V(G) and E(H) ⊆ E(G). 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 Kn [11]. The chromatic number χ(G) of a graph G is the minimum number of colors required for proper vertex coloring of G. A m−coloring of a graphG is a vertex coloring ofG that uses atmostm−colors. A graph G is said to bem-colorable if G admits a proper vertex coloring using at most m colors [11]. Let G and H be two graphs. A function φ : V(G) 󳨀→ V(H) is a homomorphism fromG toH if it preserves edges, that is, if for any edge [u, V] of G, [φ(u), φ(V)] is an edge of H [12]. A retract of a graph G is a subgraphH of G such that there exists a homomorphism r : G 󳨀→ H, called retraction with r(x) = x for any vertex x ofH [7]. A core is a graph which does not retract to a proper subgraph [12].


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][2][3][4][5][6][7][8][9][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   [11].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 [11].Let  and  be two graphs.A function  : () → () is a homomorphism from  to  if it preserves edges, that is, if for any edge [, V] of , [(), (V)] is an edge of  [12].A retract of a graph  is a subgraph  of  such that there exists a homomorphism  :  → , called retraction with () =  for any vertex  of  [7].A core is a graph which does not retract to a proper subgraph [12].

The Main Results
Aiming at our study, we will introduce the following.Definition 1.Let  1 and  2 be two connected graphs, where  1 is an edge of  1 ,  2 is an edge of  2 , and  1 ∩  2 = ; then we define the edge graph  1 ∨ 2 by gluing together the two edges  1 and  2 .
Theorem 3. The graphs  1 and  2 are subgraphs of  1 ∨ 2 .Also, for any tree  1 and  2 , the graph  1 ∨ 2 is also a tree.
Proof.The proof of this theorem is clear.

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.