On applications of bipartite graph associated with algebraic structures

Abstract The latest developments in algebra and graph theory allow us to ask a natural question, what is the application in real world of this graph associated with some mathematical system? Groups can be used to construct new non-associative algebraic structures, loops. Graph theory plays an important role in various fields through edge labeling. In this paper, we shall discuss some applications of bipartite graphs, related with Latin squares of Wilson loops, such as metabolic pathways, chemical reaction networks, routing and wavelength assignment problem, missile guidance, astronomy and x-ray crystallography.


Introduction
Ruth Moufang, German geometer, introduced Quasigroup to associate with non-desarguesian plane signicantly. Naturally, this mathematical structure is the generalization of frequently studied algebraic system, group. After the origination, mathematicians discussed it with combinatorial analysis, projective plane, experimental design, algebra, topology, etc. All algebraic nets are the examples of Quasigroups. People worked on di erent algebraic structures, initiated from magma or groupoid, in the interval 1900 to 1970 and all these developments culminated after the appearance of Moufang loops and Bol loops. Loop theory has not only history of 70 years but also moving in the direction of well-known research areas with modernity.
In the recent time researchers are using computers rapidly for mostly used applications and the second approach is graph theory. We can understand many real world applications by associating with several graphs. Graph theory is the extensively used branch of mathematics. In 1735, Koinsber bridge's problem gave the origin of graph theory and later on researchers did work on Eulerian graph, complete graph and bipartite graph comprehensively. After Leonhard Euler's work, Cauchy and L'Huilier played an important role to initiate a new branch, topology, of mathematics tremendously. Arthur Cayley was rst mathematician who used trees for chemical composition in theoretical chemistry. Sylvester used term "graph" rst time in his work and Frank Harary wrote an eminent book on graph theory in 1969 to connect mathematicians, biologists, computer experts, chemists, engineers and social scientists see Figure 1.  Graph Γ = (Σ, Υ) is known as a simple graph if it does not contain loops and multiple edges where Σ and Υ are respectively sets of vertices and edges of Γ. A simple graph Γ = (Σ, Υ) is said to be complete if there is an edge between any pair of distinct vertices. Secondly, Γ = (Σ , Σ , E) is bipartite (or 2-mode network or bigraph) if ∀ e ∈ Υ has one end in Σ and the other in Σ where the sets Σ , Σ are disjoint [ [17, p. 201] with the help of bipartite graphs. Graph theory is a companionable and proli c tool to handle chemical reaction networks (CRNs) [18, p. 2309]. Absolutely, it has become an important structure to study in di erent elds specially computer science and chemistry.
In the modern world, it seems impossible to discuss properties of classical random graphs associated with the models of real-world complex networks. Instead of classical random graphs bipartite graphs can be used to overcome this di culty [19, p. 800]. Bipartite graphs are very expedient to decode the code words in advance coding theory and Query Log Analysis, Personnel Assignment Problem, Optimal Assignment Problem. A factor graph (subclass of bipartite graph) and belief network are very closed to each other. They give us probabilistic decoding of low-density parity-check and turbo codes in [20, p. 143]. Inspired by [21, p. 332] for projective geometry, taking into account the fact that every Levi graph is the bipartite graph, we are able to model the incidences between points and lines in a con guration.
Document/Word Graphs are the bipartite graphs where (say) Σ and Σ respectively consists of documents and words, e = (v , v ) ∈ Υ represents word v is in the document v . Edge labeling of a simple graph Γ = (Σ, Υ) is a mapping, Θ : Υ → ♣, from Υ to ♣, set of integers or symbols. And with this Θ the graph Γ is called edge-labeled graph. For an healthier understanding of graph labeling, its consequences and algebraic properties see [22][23][24][25][26][27]. Without any restriction, algebraic operation, we can assign a Wilson Latin square to a complete bipartite graph through edge labeling. In Figure 2, we label an element (− , ) as an edge with respect to any two arbitrary vertices A and B so K , is desired bipartite graph for table 1 with Figure 2.   A path from u to v in the simple graph Γ is a sequence of edges (ζ , ζ ), (ζ , ζ ), (ζ , ζ ), ..., (ζ m− , ζm) in Γ, where m is a nonnegative integer, and ζ = u and ζm = v. It can be denoted by ζ , ζ , ζ , ..., ζ m− , ζm and has length m. In case of directed graphs, we say a path is increasing if the sequence of its edge labels is non-decreasing. Good edge-labeling is an edge-labeling in which for any two distinct vertices u, v we have at most one increasing (u, v)−path. Subcubic {C , K , }-free graphs, planar graphs of girth at least 6, C -free outerplanar graphs, forests are the examples of graphs which admit the good edge-labeling and help us to overcome RWA (Routing and Wavelength Assignment) problem for UPP-DAG [25,[28][29][30]. Graph labeling plays a vital role in a number of applications like data base management, communication network addressing, circuit design, x-ray crystallography, astronomy, radar and missile guidance. For further information see [31][32][33].
Which is the required identity. The converse is easy to verify.

Conclusion
This article deals with the application of graph theory in the pure mathematics. In particular the aim is to discover those algebraic structures and quasigroups which are closely associated with bipartite graphs. We have shown that graph labeling is a powerful tool to understand algebraic object namely the Wilson loop. The eld is quite open in the sense, one can discover more connections between these two areas.

Acknowledgment:
The authors are grateful to the anonymous referee for their valuable comments and suggestions that improved this paper.