# On applications of bipartite graph associated with algebraic structures

Xiujun Zhang
• Key Laboratory of Pattern Recognition and Intelligent Information Processing, Institutions of Higher Education of Sichuan Province, Chengdu University, Chengdu, 610106, China
• Email
• Search for other articles:
and Muhammad Kamran Siddiqui
• Corresponding author
• Department of Mathematics, Comsats University Islamabad, Lahore Campus, Lahore, Pakistan
• Email
• Search for other articles:

## 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.

## 1 Introduction

Ruth Moufang, German geometer, introduced Quasigroup to associate with non-desarguesian plane significantly. 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 different 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.

Let Ξ be a non-empty set such that with a binary operation ⋄, (Ξ, ⋄) is a groupoid that is ∀ α, βΞ we have αβΞ. If the system of equations pα = q and βp = q have unique solutions for α and β then (Ξ, ⋄) is known as Quasigroup. Furthermore, if there exists a unique identity element ϵΞ, then (Ξ, ⋄) is said to be a loop. For each αΞ, the elements α, αrΞ such that αα = ααr = ϵ are called left and right inverses of α respectively. Ξ is known as Wilson loop (WL) if and only if it obeys the Wilson Identity (WI);

$α⋄(α⋄β)r=(α⋄γ)(α⋄(β⋄γ))r,∀α,β,γ∈Ξ$

equivalently;

$(β⋄α)ℓ⋄α=((γ⋄β)⋄α)ℓ⋄(γ⋄α),∀α,β,γ∈Ξ$

Any loop Ξ satisfying α⋄(βα) = (αβ)⋄α is flexible loopα, βΞ. Sets ℵ = {pΞ; p ⋄ (αβ) = (pα) ⋄ βα, βΞ}, ℵχ = {pΞ; α ⋄ (pβ) = (αp) ⋄ βα, βΞ} and ℵr = {pΞ; (αβ) ⋄ p = α ⋄ (βp) ∀ α, βΞ} are said to be left, middle and right nucleus, respectively. The set ℵ = ℵ ∩ ℵχ ∩ ℵr consists of all elements that associate with any other two elements and is called the nucleus of Ξ. For Wilson loop we have ℵ = ℵ = ℵχ = ℵr. Ξ is weak inverse property loop if and only if (αβ) ⋄ γ = ϵ implies α ⋄ (βγ) = ϵα, β, γΞ. Ξ is called conjugacy closed loop if the sets of left and right translations are closed under conjugation.

As worked by Goodaire and Robinson [1, Theorem 1], a loop Ξ is a Wilson loop iff it is weak inverse property loop [2, p. 295][3, p. 132] and conjugacy closed loop [4, p. 843]. Originally Wilson loop is introduced by E. L. Wilson in [5, Theorem 5] where it is also given that a Moufang loop [6, p. 42][7, p. 194] is Wilson loop if and only if α2 ∈ ℵ ∀ αΞ. The developments of loop theory remained eclipsed under the fast moving research horizon of the theory of groups. After the completion of the list of simple groups, the research environment is getting more suitability for the structures of non-associative models like loops and Quasigroups. In the literature of loop theory, the groups are being used to derive new families of loops.

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 first mathematician who used trees for chemical composition in theoretical chemistry. Sylvester used term “graph” first 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, Y) is known as a simple graph if it does not contain loops and multiple edges where Σ and Y are respectively sets of vertices and edges of Γ. A simple graph Γ = (Σ, Y) is said to be complete if there is an edge between any pair of distinct vertices. Secondly, Γ = (Σ1, Σ2, E) is bipartite (or 2-mode network or bigraph) if ∀ eY has one end in Σ1 and the other in Σ2 where the sets Σ1, Σ2 are disjoint [8, p. 2][9, p. 225]. Equivalently, Γ is bipartite if it does not contain any odd length cycle. For instance, Kn,n is a complete bipartite graph with cardinality of both Σ1, Σ2 is n. Graph, Γ = (Σ1, Σ2, Y) is balanced bipartite graph if |Σ1| = |Σ2|. Bipartite graphs can be used broadly to consider bioentities, signal transduction, gene regulation, evolutionary relationships, metabolic pathways, gene expression etc. as vertices and their correlation as edges within a network.

Now biologists can understand more about yeast-two-hybrid [10, p. 246], protein-protein interactions (PPIs) for particular organisms [11, p. 822][12, p. 4570][13, p. 4880][14, p. 212][15, p. 624]. Microarrays and RNA-seq [16, p. 57][17, p. 201] with the help of bipartite graphs. Graph theory is a companionable and prolific tool to handle chemical reaction networks (CRNs) [18, p. 2309]. Absolutely, it has become an important structure to study in different fields 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 difficulty [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 configuration.

Document/Word Graphs are the bipartite graphs where (say) Σ1 and Σ2 respectively consists of documents and words, e = (v1, v2) ∈ Y represents word v2 is in the document v1. Edge labeling of a simple graph Γ = (Σ, Y) is a mapping, Θ: Y → ♣, from Y 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 (−1, 1) as an edge with respect to any two arbitrary vertices A and B so K4,4 is desired bipartite graph for table 1 with Figure 2.

Table 1

Wilson loop of order 4.

 (1, 0) (1, 1) (−1, 0) (−1, 1) (1, 1) (1, 0) (−1, 1) (−1, 0) (−1, 0) (−1, 1) (1, 1) (1, 0) (−1, 1) (−1, 0) (1, 0) (1, 1)

A path from u to v in the simple graph Γ is a sequence of edges (ζ0, ζ1), (ζ1, ζ2), (ζ2, ζ3), …, (ζm−1, ζm) in Γ, where m is a nonnegative integer, and ζ0 = u and ζm = v. It can be denoted by ζ0, ζ1, ζ2, …, ζm−1, 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 {C3, K2,3}-free graphs, planar graphs of girth at least 6, C3-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].

## 2 Main results

Let Ψ1 and Ψ2 be respectively groups under multiplication and addition. Moreover Ψ2 is abelian group. The function ♭: Ψ1 × Ψ1Ψ2 with;

$(1,ϝ)♭=(ϝ,1)♭=0,∀ϝ∈Ψ1,$

is called a factor set. Binary operation ⋄ on Ψ1 × Ψ2 can be defined, with the help of ♭, as follows;

$(ϝ1,ν1)⋄(ϝ2,ν2)=(ϝ1ϝ2,ν1+ν2+(ϝ1,ϝ2)♭)∀ϝ1,ϝ2∈Ψ1 and ∀ν1,ν2∈Ψ2.$

Clearly the resulting groupoid is a loop denoted by (Ψ1, Ψ2, ♭) with neutral element (1, 0). Note that (ϝ, ν)−1 = (ϝ−1, − ν − (ϝ, ϝ−1)♭) is the inverse of (ϝ, ν) in (Ψ1, Ψ2, ♭). The following theorem provides construction of the Wilson loops.

Theorem 1

Let ♭ : Ψ1 × Ψ1Ψ2be a factor set. Then (Ψ1, Ψ2, ♭) is Wilson loop if and only if

$(ϝ1,ϝ3)♭+(ϝ1,ϝ2)♭+(ϝ1ϝ2,(ϝ1ϝ2)−1)♭+(ϝ1ϝ3,(ϝ1⋅ϝ2ϝ3)−1)♭=(ϝ2,ϝ3)♭+(ϝ1,ϝ2ϝ3)♭+(ϝ1⋅ϝ2ϝ3,(ϝ1⋅ϝ2ϝ3)−1)♭+(ϝ1,(ϝ1ϝ2)−1)♭.$

Proof

Let (Ψ1, Ψ2, ♭) is the Wilson loop so it satisfies the identity

$(ϝ1,ν1)⋄((ϝ1,ν1)⋄(ϝ2,ν2))r=((ϝ1,ν1)⋄(ϝ3,ν3))⋄((ϝ1,ν1)⋄((ϝ2,ν2)⋄(ϝ3,ν3)))r$

for all (ϝ1, ν1), (ϝ2, ν2), (ϝ3, ν3) ∈ (Ψ1, Ψ2, ♭).

Now

$(ϝ1,ν1)⋄((ϝ1,ν1)⋄(ϝ2,ν2))r=(ϝ1,ν1)⋄((ϝ1ϝ2,ν1+ν2+(ϝ1,ϝ2)♭)r=(ϝ1,ν1)⋄((ϝ1ϝ2)−1,−ν1−ν2−(ϝ1,ϝ2)♭−(ϝ1ϝ2,(ϝ1ϝ2)−1)♭)=(ϝ1,ν1)⋄(ϝ2−1ϝ1−1,−ν1−ν2−(ϝ1,ϝ2)♭−(ϝ1ϝ2,ϝ2−1ϝ1−1)♭)=(ϝ1⋅ϝ2−1ϝ1−1,ν1−ν1−ν2−(ϝ1,ϝ2)♭−(ϝ1ϝ2,ϝ2−1ϝ1−1)♭+(ϝ1,ϝ2−1ϝ1−1)♭)=(ϝ1⋅ϝ2−1ϝ1−1,−ν2−(ϝ1,ϝ2)♭−(ϝ1ϝ2,ϝ2−1ϝ1−1)♭+(ϝ1,ϝ2−1ϝ1−1)♭)$

and

$((ϝ1,ν1)⋄(ϝ3,ν3))⋄((ϝ1,ν1)⋄((ϝ2,ν2)⋄(ϝ3,ν3)))r=(ϝ1ϝ3,ν1+ν3+(ϝ1,ϝ3)♭)⋄((ϝ1,ν1)⋄(ϝ2ϝ3,ν2+ν3+(ϝ2,ϝ3)♭))r=(ϝ1ϝ3,ν1+ν3+(ϝ1,ϝ3)♭)⋄((ϝ1⋅ϝ2ϝ3,ν1+ν2+ν3+(ϝ2,ϝ3)♭+(ϝ1,ϝ2ϝ3)♭)r=(ϝ1ϝ3,ν1+ν3+(ϝ1,ϝ3)♭)⋄((ϝ1⋅ϝ2ϝ3)−1,−ν1−ν2−ν3−(ϝ2,ϝ3)♭ −(ϝ1,ϝ2ϝ3)♭−(ϝ1⋅ϝ2ϝ3,(ϝ1⋅ϝ2ϝ3)−1)♭=(ϝ1ϝ3,ν1+ν3+(ϝ1,ϝ3)♭)⋄((ϝ3−1ϝ2−1⋅ϝ1−1,−ν1−ν2−ν3−(ϝ2,ϝ3)♭−(ϝ1,ϝ2ϝ3)♭ −(ϝ1⋅ϝ2ϝ3,ϝ3−1ϝ2−1⋅ϝ1−1)♭)=((ϝ1ϝ3)(ϝ3−1ϝ2−1⋅ϝ1−1),ν1+ν3+(ϝ1,ϝ3)♭−ν1−ν2−ν3−(ϝ2,ϝ3)♭−(ϝ1,ϝ2ϝ3)♭ −(ϝ1⋅ϝ2ϝ3,ϝ3−1ϝ2−1⋅ϝ1−1)♭)=(ϝ1⋅ϝ2−1ϝ1−1,(ϝ1,ϝ3)♭−(ϝ2,ϝ3)♭−(ϝ1,ϝ2ϝ3)♭−(ϝ1⋅ϝ2ϝ3,ϝ3−1ϝ2−1⋅ϝ1−1)♭ −ν2+(ϝ1ϝ3,ϝ3−1ϝ2−1⋅ϝ1−1)♭)$

Using both results in the above Wilson identity

$(ϝ1,ϝ3)♭+(ϝ1,ϝ2)♭+(ϝ1ϝ2,(ϝ1ϝ2)−1)♭+(ϝ1ϝ3,(ϝ1⋅ϝ2ϝ3)−1)♭=(ϝ2,ϝ3)♭+(ϝ1,ϝ2ϝ3)♭+(ϝ1⋅ϝ2ϝ3,(ϝ1⋅ϝ2ϝ3)−1)♭+(ϝ1,(ϝ1ϝ2)−1)♭$

Which is the required identity. The converse is easy to verify. □

### 2.1 Wilson factor set

A factor set ♭ : Ψ1 × Ψ1Ψ2 with equation (1) is called Wilson factor-set. If |Ψ1| = 2s where s is the whole number then equation (1) reduces to

$(ϝ1,ϝ3)♭+(ϝ1,ϝ2)♭+(ϝ1ϝ2,ϝ1ϝ2)♭+(ϝ1ϝ3,ϝ1⋅ϝ2ϝ3)♭=(ϝ2,ϝ3)♭+(ϝ1,ϝ2ϝ3)♭+(ϝ1⋅ϝ2ϝ3,ϝ1⋅ϝ2ϝ3)♭+(ϝ1,ϝ1ϝ2)♭.$

This Wilson-factor set is very helpful in construction of Wilson loops by the following manner.

Proposition 1

LetΨ2be an additive abelian group with cardinalityk, positive integer greater than 1, and 0 ≠ pΨ2. LetΨ1 = {1, ℶ} be the multiplicative group where ℶ = cosπ + ιsinπ. We define function ♭:Ψ1 × Ψ1Ψ2by

$(ϝ1,ϝ2)♭=p,if(ϝ1,ϝ2)=(ℶ,ℶ);0,if(ϝ1,ϝ2)=(1,1),(1,ℶ),(ℶ,1).$

Then (Ψ1, Ψ2, ♭) is a flexible, non-associative Wilson loop with nucleus ℵ = (1, ν) ∀ νΨ2and ∀ ϝ1, ϝ2Ψ1.

Let Ψ1 = {1, ℶ}, multiplicative group, and Ψ2 = {0, 1, 2, 3, …, n − 1}, additive abelian group of modulo n, table 2 shows a pattern of Wilson loops of even orders.

Table 2

Wilson loop of order 2n.

(1, 0)(1, 1)(1, 2)(1, 3)(1, n − 2)(1, n − 1)(ℶ, 0)(ℶ, 1)(ℶ, 2)(ℶ, 3)(ℶ, n − 2)(ℶ, n − 1)
(1, 0)(1, 0)(1, 1)(1, 2)(1, 3)(1, n − 2)(1, n − 1)(ℶ, 0)(ℶ, 1)(ℶ, 2)(ℶ, 3)(ℶ, n − 2)(ℶ, n − 1)
(1, 1)(1, 1)(1, 2)(1, 3)(1, 4)(1, n − 1)(1, 0)(ℶ, 1)(ℶ, 2)(ℶ, 3)(ℶ, 4)(ℶ, n − 1)(ℶ, 0)
(1, 2)(1, 2)(1, 3)(1, 4)(1, 5)(1, 0)(1, 1)(ℶ, 2)(ℶ, 3)(ℶ, 4)(ℶ, 5)(ℶ, 0)(ℶ, 1)
(1, 3)(1, 3)(1, 4)(1, 5)(1, 6)(1, 1)(1, 2)(ℶ, 3)(ℶ, 4)(ℶ, 5)(ℶ, 6)(ℶ, 1)(ℶ, 2)
.............
.............
.............
(1, n − 2)(1, n − 2)(1, n − 1)(1, 0)(1, 1)(1, n − 4)(1, n − 3)(ℶ, n − 2)(ℶ, n − 1)(ℶ, 0)(ℶ, 1)(ℶ, n − 4)(ℶ, n − 3)
(1, n − 1)(1, n − 1)(1, 0)(1, 1)(1, 2)(1, n − 3)(1, n − 2)(ℶ, n − 1)(ℶ, 0)(ℶ, 1)(ℶ, 2)(ℶ, n − 3)(ℶ, n − 2)
(ℶ, 0)(ℶ, 0)(ℶ, 1)(ℶ, 2)(ℶ, 3)(ℶ, n − 2)(ℶ, n − 1)(1, n − 1)(1, 0)(1, 1)(1, 2)(1, n − 3)(1, n − 2)
(ℶ, 1)(ℶ, 1)(ℶ, 2)(ℶ, 3)(ℶ, 4)(ℶ, n − 1)(ℶ, 0)(1, 0)(1, 1)(1, 2)(1, 3)(1, n − 2)(1, n − 1)
(ℶ, 2)(ℶ, 2)(ℶ, 3)(ℶ, 4)(ℶ, 5)(ℶ, 0)(ℶ, 1)(1, 1)(1, 2)(1, 3)(1, 4)(1, n − 1)(1, 0)
(ℶ, 3)(ℶ, 3)(ℶ, 4)(ℶ, 5)(ℶ, 6)(ℶ, 1)(ℶ, 2)(1, 2)(1, 3)(1, 4)(1, 5)(1, 0)(1, 1)
.............
.............
.............
(ℶ, n − 2)(ℶ, n − 2)(ℶ, n − 1)(ℶ, 0)(ℶ, 1)(ℶ, n − 4)(ℶ, n − 3)(1, n − 3)(1, n − 2)(1, n − 1)(1, 0)(1, n − 5)(1, n − 4)
(ℶ, n − 1)(ℶ, n − 1)(ℶ, 0)(ℶ, 1)(ℶ, 2)(ℶ, n − 3)(ℶ, n − 2)(1, n − 2)(1, n − 1)(1, 0)(1, 1)(1, n − 4)(1, n − 3)

Proposition 2

LetΨ2be an additive abelian group with |Ψ2| > 2, and order ofpis greater than 2 where 0 ≠ pΨ2. LetΨ1 = {1, 1, 2, 3} be the Klein group. Define ♭ : Ψ1 × Ψ1Ψ2by

$(ϝ1,ϝ2)♭=p,if (ϝ1,ϝ2)=(℘1,℘3),(℘3,℘2),(℘2,℘1);−p, if (ϝ1,ϝ2)=(℘1,℘2),(℘2,℘3),(℘3,℘1);0,otherwise.$

Then (Ψ1, Ψ2, ♭) is a non-flexible (implies non-associative) Wilson loop with nucleus ℵ = (1, ν) ∀ νΨ2and ∀ ϝ1, ϝ2Ψ1.

Proof

Following table shows that function ♭ is obviously Wilson-factor set.

$♭1℘1℘2℘310000℘100−pp℘20p0−p℘30−pp0$

To show that (Ψ1, Ψ2, ♭) is Wilson loop we verify equation (2). Since ♭ is factor set, there is nothing to prove when ϝ3 = 1;

$(ϝ1,ϝ2)♭+(ϝ1ϝ2,ϝ1ϝ2)♭+(ϝ1,ϝ1ϝ2)♭=(ϝ1,ϝ2)♭+(ϝ1ϝ2,ϝ1ϝ2)♭+(ϝ1,ϝ1ϝ2)♭.$

When ϝ2 = 1;

$(ϝ1,ϝ3)♭+(ϝ1,ϝ1)♭+(ϝ1ϝ3,ϝ1ϝ3)♭=(ϝ1,ϝ3)♭+(ϝ1ϝ3,ϝ1ϝ3)♭+(ϝ1,ϝ1)♭.$

Similarly it can be proved for ϝ1 = 1.

When ϝ3 = 1;

$(ϝ1,℘1)♭+(ϝ1,ϝ2)♭+(ϝ1ϝ2,ϝ1ϝ2)♭+(ϝ1℘1,ϝ1ϝ2⋅℘1)♭=(ϝ2,℘1)♭+(ϝ1,ϝ2℘1)♭+(ϝ1⋅ϝ2℘1,ϝ1⋅ϝ2℘1)♭+(ϝ1,ϝ1ϝ2)♭.$

Putting ϝ1 = 2, ϝ2 = 3 in the last identity, we have

$(ϝ1,℘1)♭+(ϝ1,ϝ2)♭+(ϝ1ϝ2,ϝ1ϝ2)♭+(ϝ1℘1,ϝ1ϝ2⋅℘1)♭=(℘2,℘1)♭+(℘2,℘3)♭+(℘2℘3,℘2℘3)♭+(℘2℘1,℘2⋅℘3℘1)♭=p−p+0+(℘2℘1,℘2⋅℘3℘1)♭=(℘2℘1,℘2⋅℘3℘1)♭=(℘3,1)♭$

$(ϝ2,℘1)♭+(ϝ1,ϝ2℘1)♭+(ϝ1⋅ϝ2℘1,ϝ1⋅ϝ2℘1)♭+(ϝ1,ϝ1ϝ2)♭=(℘3,℘1)♭+(℘2,℘3℘1)♭+(℘2⋅℘3℘1,℘2⋅℘3℘1)♭+(℘2,℘2℘3)♭=−p+(℘2,℘3℘1)♭+(℘2,℘2℘3)♭=−p+(℘2,℘2)♭+(℘2,℘1)♭$

Similarly we can check other cases when ϝ3 = 1. By using same procedure for ϝ2, ϝ1 we can verify (2). Thus (Ψ1, Ψ2, ♭) is Wilson loop. (Ψ1, Ψ2, ♭) is non-commutative, non-associative Wilson loop. As let ∀ νΨ2 and 0 ≠ p, p ≠ − p

$((℘1,ν)⋄(℘2,ν))⋄(℘1,ν)=(℘1℘2,ν+ν+(℘1,℘2)♭)⋄(℘1,ν)=(℘3,2ν−p)⋄(℘1,ν)=(℘3℘1,2ν−p+ν+(℘3,℘1)♭)=(℘2,3ν−2p)$

$(℘1,ν)⋄((℘2,ν)⋄(℘1,ν))=(℘1,ν)⋄(℘2℘1,ν+ν+(℘2,℘1)♭)=(℘1,ν)⋄(℘3,2ν+p)=(℘2,3ν+p+(℘1,℘3)♭)=(℘2,3ν+2p)$

from (3) and (4)

$((℘1,ν)⋄(℘2,ν))⋄(℘1,ν)≠(℘1,ν)⋄((℘2,ν)⋄(℘1,ν)).$

It implies that (Ψ1, Ψ2, ♭) is not flexible and (1, ν), (2, ν) are not in ℵ. Similarly ((1, ν) ⋄ (3, ν)) ⋄ (1, ν) ≠ ((1, ν) ⋄ (3, ν)) ⋄ (1, ν) gives (3, ν) also not in ℵ. Finally ∀ ϝ1, ϝ2Ψ1 and ∀ ν2, ν3Ψ2

$((1,ν)⋄(ϝ1,ν2))⋄(ϝ2,ν3)=(ϝ1ϝ2,ν+ν2+ν3+(ϝ1,ϝ2)♭)=(1,ν)⋄((ϝ1,ν2)⋄(ϝ2,ν3))$

shows that (1, ν) ∈ ℵ represents star graph through above mentioned edge labeling. □

Example

If Ψ1 = {1, 1, 2, 3} and Ψ2 = {0, 1, 2, 3, 4}, with modulo 5, then K20,20 is the associated graph see Figure 3. Let Ψ1 = {1, 1, 2, 3}, Klein four group, and Ψ2 = {0, 1, 2, 3, …, n − 1}, additive abelian group of modulo n, table 3 also shows a pattern of Wilson loops.

Table 3

Wilson loop of order 4n.

(1, 0)(1, 1)(1, 2)(1, 3)(1, n − 2)(1, n − 1)(1, 0)(1, 1)(1, 2)(1, 3)(1, n − 2)(1, n − 1)(2, 0)(2, 1)(2, 2)(2, 3)(2, n − 2)(2, n − 1)(3, 0)(3, 1)(3, 2)(3, 3)(3, n − 2)(3, n − 1)
(1, 0)(1, 0)(1, 1)(1, 2)(1, 3)(1, n − 2)(1, n − 1)(1, 0)(1, 1)(1, 2)(1, 3)(1, n − 2)(1, n − 1)(2, 0)(2, 1)(2, 2)(2, 3)(2, n − 2)(2, n − 1)(3, 0)(3, 1)(3, 2)(3, 3)(3, n − 2)(3, n − 1)
(1, 1)(1, 1)(1, 2)(1, 3)(1, 4)(1, n − 1)(1, 0)(1, 1)(1, 2)(1, 3)(1, 4)(1, n − 1)(1, 0)(2, 1)(2, 2)(2, 3)(2, 4)(2, n − 1)(2, 0)(3, 1)(3, 2)(3, 3)(3, 4)(3, n − 1)(3, 0)
(1, 2)(1, 2)(1, 3)(1, 4)(1, 5)(1, 0)(1, 1)(1, 2)(1, 3)(1, 4)(1, 5)(1, 0)(1, 1)(2, 2)(2, 3)(2, 4)(2, 5)(2, 0)(2, 1)(3, 2)(3, 3)(3, 4)(3, 5)(3, 0)(3, 1)
(1, 3)(1, 3)(1, 4)(1, 5)(1, 6)(1, 1)(1, 2)(1, 3)(1, 4)(1, 5)(1, 6)(1, 1)(1, 2)(2, 3)(2, 4)(2, 5)(2, 6)(2, 1)(2, 2)(3, 3)(3, 4)(3, 5)(3, 6)(3, 1)(3, 2)
.........................
.........................
.........................
(1, n − 2)(1, n − 2)(1, n − 1)(1, 0)(1, 1)(1, n − 4)(1, n − 3)(1, n − 2)(1, n − 1)(1, 0)(1, 1)(1, n − 4)(1, n − 3)(2, n − 2)(2, n − 1)(2, 0)(2, 1)(2, n − 4)(2, n − 3)(3, n − 2)(3, n − 1)(3, 0)(3, 1)(3, n − 4)(3, n − 3)
(1, n − 1)(1, n − 1)(1, 0)(1, 1)(1, 2)(1, n − 3)(1, n − 2)(1, n − 1)(1, 0)(1, 1)(1, 2)(1, n − 3)(1, n − 2)(2, n − 1)(2, 0)(2, 1)(2, 2)(2, n − 3)(2, n − 2)(3, n − 1)(3, 0)(3, 1)(3, 2)(3, n − 3)(3, n − 2)
(1, 0)(1, 0)(1, 1)(1, 2)(1, 3)(1, n − 2)(1, n − 1)(1, 0)(1, 1)(1, 2)(1, 3)(1, n − 2)(1, n − 1)(3, 1)(3, 2)(3, 3)(3, 4)(3, n − 1)(3, 0)(2, n − 1)(2, 0)(2, 1)(2, 2)(2, n − 3)(2, n − 2)
(1, 1)(1, 1)(1, 2)(1, 3)(1, 4)(1, n − 1)(1, 0)(1, 1)(1, 2)(1, 3)(1, 4)(1, n − 1)(1, 0)(3, 2)(3, 3)(3, 4)(3, 5)(3, 0)(3, 1)(2, 0)(2, 1)(2, 2)(2, 3)(2, n − 2)(2, n − 1)
(1, 2)(1, 2)(1, 3)(1, 4)(1, 5)(1, 0)(1, 1)(1, 2)(1, 3)(1, 4)(1, 5)(1, 0)(1, 1)(3, 3)(3, 4)(3, 5)(3, 6)(3, 1)(3, 2)(2, 1)(2, 2)(2, 3)(2, 4)(2, n − 1)(2, 0)
(1, 3)(1, 3)(1, 4)(1, 5)(1, 6)(1, 1)(1, 2)(1, 3)(1, 4)(1, 5)(1, 6)(1, 1)(1, 2)(3, 4)(3, 5)(3, 6)(3, 7)(3, 2)(3, 3)(2, 2)(2, 3)(2, 4)(2, 5)(2, 0)(2, 1)
.........................
.........................
.........................
(1, n − 2)(1, n − 2)(1, n − 1)(1, 0)(1, 1)(1, n − 4)(1, n − 3)(1, n − 2)(1, n − 1)(1, 0)(1, 1)(1, n − 4)(1, n − 3)(3, n − 1)(3, 0)(3, 1)(3, 2)(3, n − 3)(3, n − 2)(2, n − 3)(2, n − 2)(2, n − 1)(2, 0)(2, n − 5)(2, n − 4)
(1, n − 1)(1, n − 1)(1, 0)(1, 1)(1, 2)(1, n − 3)(1, n − 2)(1, n − 1)(1, 0)(1, 1)(1, 2)(1, n − 3)(1, n − 2)(3, 0)(3, 1)(3, 2)(3, 3)(3, n − 2)(3, n − 1)(2, n − 2)(2, n − 1)(2, 0)(2, 1)(2, n − 4)(2, n − 3)
(2, 0)(2, 0)(2, 1)(2, 2)(2, 3)(2, n − 2)(2, n − 1)(3, n − 1)(3, 0)(3, 1)(3, 2)(3, n − 3)(3, n − 2)(1, 0)(1, 1)(1, 2)(1, 3)(1, n − 2)(1, n − 1)(1, 1)(1, 2)(1, 3)(1, 4) (1, n − 1)(1, 0)
(2, 1)(2, 1)(2, 2)(2, 3)(2, 4)(2, n − 1)(2, 0)(3, 0)(3, 1)(3, 2)(3, 3)(3, n − 2)(3, n − 1)(1, 1)(1, 2)(1, 3)(1, 4)(1, n − 1)(1, 0)(1, 2)(1, 3)(1, 4)(1, 5)(1, 0)(1, 1)
(2, 2)(2, 2)(2, 3)(2, 4)(2, 5)(2, 0)(2, 1)(3, 1)(3, 2)(3, 3)(3, 4)(3, n − 1)(3, 0)(1, 2)(1, 3)(1, 4)(1, 5)(1, 0)(1, 1)(1, 3)(1, 4)(1, 5)(1, 6)(1, 1)(1, 2)
(2, 3)(2, 3)(2, 4)(2, 5)(2, 6)(2, 1)(2, 2)(3, 2)(3, 3)(3, 4)(3, 5)(3, 0)(3, 1)(1, 3)(1, 4)(1, 5)(1, 6)(1, 1)(1, 2)(1, 4)(1, 5)(1, 6)(1, 7)(1, 2)(1, 3)
.........................
.........................
.........................
(2, n − 2)(2, n − 2)(2, n − 1)(2, 0)(2, 1)(2, n − 4)(2, n − 3)(3, n − 3)(3, n − 2)(3, n − 1)(3, 0)(3, n − 5)(3, n − 4)(1, n − 2)(1, n − 1)(1, 0)(1, 1)(1, n − 4)(1, n − 3)(1, n − 1)(1, 0)(1, 1)(1, 2)(1, n − 3)(1, n − 2)
(2, n − 1)(2, n − 1)(2, 0)(2, 1)(2, 2)(2, n − 3)(2, n − 2)(3, n − 2)(3, n − 1)(3, 0)(3, 1)(3, n − 4)(3, n − 3)(1, n − 1)(1, 0)(1, 1)(1, 2)(1, n − 3)(1, n − 2)(1, 0)(1, 1)(1, 2)(1, 3)(1, n − 2)(1, n − 1)
(3, 0)(3, 0)(3, 1)(3, 2)(3, 3)(3, n − 2)(3, n − 1)(2, 1)(2, 2)(2, 3)(2, 4)(2, n − 1)(2, 0)(1, n − 1)(1, 0)(1, 1)(1, 2)(1, n − 3)(1, n − 2)(1, 0)(1, 1)(1, 2)(1, 3)(1, n − 2)(1, n − 1)
(3, 1)(3, 1)(3, 2)(3, 3)(3, 4)(3, n − 1)(3, 0)(2, 2)(2, 3)(2, 4)(2, 5)(2, 0)(2, 1)(1, 0)(1, 1)(1, 2)(1, 3)(1, n − 2)(1, n − 1)(1, 1)(1, 2)(1, 3)(1, 4)(1, n − 1)(1, 0)
(3, 2)(3, 2)(3, 3)(3, 4)(3, 5)(3, 0)(3, 1)(2, 3)(2, 4)(2, 5)(2, 6)(2, 1)(2, 2)(1, 1)(1, 2)(1, 3)(1, 4)(1, n − 1)(1, 0)(1, 2)(1, 3)(1, 4)(1, 5)(1, 0)(1, 1)
(3, 3)(3, 3)(3, 4)(3, 5)(3, 6)(3, 1)(3, 2)(2, 4)(2, 5)(2, 6)(2, 7)(2, 2)(2, 3)(1, 2)(1, 3)(1, 4)(1, 5)(1, 0)(1, 1)(1, 3)(1, 4)(1, 5)(1, 6)(1, 1)(1, 2)
.........................
.........................
.........................
(3, n − 2)(3, n − 2)(3, n − 1)(3, 0)(3, 1)(3, n − 4)(3, n − 3)(2, n − 1)(2, 0)(2, 1)(2, 2)(2, n − 3)(2, n − 2)(1, n − 3)(1, n − 2)(1, n − 1)(1, 0)(1, n − 5)(1, n − 4)(1, n − 2)(1, n − 1)(1, 0)(1, 1)(1, n − 4)(1, n − 3)
(3, n − 1)(3, n − 1)(3, 0)(3, 1)(3, 2)(3, n − 3)(3, n − 2)(2, 0)(2, 1)(2, 2)(2, 3)(2, n − 2)(2, n − 1)(1, n − 2)(1, n − 1)(1, 0)(1, 1)(1, n − 4)(1, n − 3)(1, n − 1)(1, 0)(1, 1)(1, 2)(1, n − 3)(1, n − 2)

We can recapitulate all the above discussion in the table 4.

Table 4

Complete bipartite graphs associated with loops.

Multiplicative GroupAdditive groupLoopppBipartite graphStar graph
Ψ1 = {1, ℶ}Ψ2 = {0, 1}(Ψ1, Ψ2, ♭)11K4,4K1,2
Ψ1 = {1, ℶ}Ψ2 = {0, 1, 2}(Ψ1, Ψ2, ♭)21K6,6K1,3
Ψ1 = {1, ℶ}Ψ2 = {0, 1, 2, 3}(Ψ1, Ψ2, ♭)31K8,8K1,4
Ψ1 = {1, ℶ}Ψ2 = {0, 1, 2, 3, …, n − 1}(Ψ1, Ψ2, ♭)n − 11K2n,2nK1,n
Ψ1 = {1, 1, 2, 3}Ψ2 = {0, 1, 2}(Ψ1, Ψ2, ♭)21K12,12K1,3
Ψ1 = {1, 1, 2, 3}Ψ2 = {0, 1, 2, 3}(Ψ1, Ψ2, ♭)31K16,16K1,4
Ψ1 = {1, 1, 2, 3}Ψ2 = {0, 1, 2, 3, 4}(Ψ1, Ψ2, ♭)41K20,20K1,5
Ψ1 = {1, 1, 2, 3}Ψ2 = {0, 1, 2, 3, …, n-1}(Ψ1, Ψ2, ♭)n − 11K4n,4nK1,n

## 3 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 field 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.

This research was supported by the Applied Basic Research (Key Project) of Sichuan Province under grant 2017JY0095 and the Soft Science Project of Sichuan Province under grant 2017ZR0041. Also this research is supported by Higher Education Commission of Pakistan under NRPU project “Properties of Ranking Ideals” via Grant No.20-3665/R&D/HEC/14 /699.

## References

• [1]

E.G. Goodaire and D.A. Robinson, Some special conjugacy closed loops, Canad. Math. Bull. 33 (1990), no. 1, 73–78, .

• Crossref
• Export Citation
• [2]

M.J. Osborn, Loops with the weak inverse property, Pacific J. Math. 10 (1960), no. 1, 295–304, .

• Crossref
• Export Citation
• [3]

R. Artzy, Inverse-cycles in weak-inverse loops, Proc. Amer. Math. Soc. 68 (1978), no. 2, 132–134, .

• Crossref
• Export Citation
• [4]

A. Drapal, Conjugacy closed loops and their multiplication groups, J. Algebra 272 (2004), no. 2, 838–850, .

• Crossref
• Export Citation
• [5]

E.L. Wilson, A class of loops with the isotopy-isomorphy property, Canad. J. Math. 18 (1966), 589–592, .

• Crossref
• Export Citation
• [6]

A. Grishkov and V.A. Zavarnitsine, Lagrange’s theorem for Moufang loops, Math. Proc. Cambridge Philos. Soc. 139 (2005), no. 1, 41–57, .

• Crossref
• Export Citation
• [7]

P.G. Nagy and P. Vojtěchovský, Automorphism groups of simple Moufang loops over perfect fields, Math. Proc. Cambridge Philos. Soc. 135 (2003), no. 2, 193–197, .

• Crossref
• Export Citation
• [8]

R.J. Shajila and S. Vimala, Graceful labelling for complete bipartite fuzzy graphs, Br. J. Math. Comput. Sci. 22 (2017), no. 2, Article no. BJMCS.32242, .

• Crossref
• Export Citation
• [9]

J.E. Hopcroft and R.M. Karp, An n 5/2 algorithm for maximum matchings in bipartite graphs, SIAM J. Comput. 2 (1973), 225–231, .

• Crossref
• Export Citation
• [10]

S. Fields and O. Song, A novel genetic system to detect protein-protein interactions, Nature 340 (1989), 245–246, .

• Crossref
• Export Citation
• [11]

A. Davy, P. Bello, N. Thierry-Mieg, P. Vaglio, J. Hitti, L. Doucette-Stamm, et al., A protein-protein interaction map of the Caenorhabditis elegans $26S$ proteasome, EMBO Rep. 2 (2001), no. 9, 821–828, .

• Crossref
• Export Citation
• [12]

T. Ito, T. Chiba, R. Ozawa, M. Yoshida, M. Hattori, and Y. Sakaki, A comprehensive two-hybrid analysis to explore the yeast protein interactome, Proc. Natl. Acad. Sci. USA 98 (2001), no. 8, 4569–4574, .

• Crossref
• Export Citation
• [13]

S. McCraith, T. Holtzman, B. Moss, and S. Fields, Genome-wide analysis of vaccinia virus protein-protein interactions, Proc. Natl. Acad. Sci. USA 97 (2000), no. 9, 4879–4884, .

• Crossref
• Export Citation
• [14]

J.C. Rain, L. Selig, H. de Reuse, V. Battaglia, C. Reverdy, S. Simon, et al., The protein-protein interaction map of Helicobacter pylori, Nature 409 (2001), 211–215, .

• Crossref
• Export Citation
• [15]

X. Shi, M.K. Hassanzadeh-Aghdam, and R. Ansari, Viscoelastic analysis of silica nanoparticle-polymer nanocomposites, Composites Part B: Eng. 158 (2019), 169–178, .

• Crossref
• Export Citation
• [16]

Z. Wang, M. Gerstein, and M. Snyder, RNA-Seq: a revolutionary tool for transcriptomics, Nat. Rev. Genet. 10 (2009), no. 1, 57–63, .

• Crossref
• Export Citation
• [17]

D. Yu, M. Kim, G. Xiao, and T.H. Hwang, Review of biological network data and its applications, Genomics Inform. 11 (2013), no. 4, 200–210, .

• Crossref
• Export Citation
• [18]

O. Sinanoglu, Theory of chemical reaction networks. All possible mechanisms or synthetic pathways with given number of reaction steps or species, J. Am. Chem. Soc. 97 (1975), no. 9, 2309–2320, .

• Crossref
• Export Citation
• [19]

J.L. Guillaume and M. Latapy, Bipartite graphs as models of complex networks, Phys. A 371 (2006), no. 2, 795–813, .

• Crossref
• Export Citation
• [20]

M.S. Sadi, P.K. Bachar, P. Ghosh, and M.S. Rahman, A new error correction coding approach, J. Adv. Inf. Technol. 4 (2013), no. 3, 142–147, .

• Crossref
• Export Citation
• [21]

A. Betten, G. Brinkmann, and T. Pisanski, Counting symmetric configurations V 3, Discrete Appl. Math. 99 (2000), no. 1-3, 331–338, .

• Crossref
• Export Citation
• [22]

Z. Shao, M.K. Siddiqui, and M.H. Muhammad, Computing Zagreb indices and Zagreb polynomials for symmetrical nanotubes, Symmetry 10 (2018), no. 7, 244, .

• Crossref
• Export Citation
• [23]

J.A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. #DS6 (2010), 246 pages, Available: http://www.combinatorics.org.

• [24]

Z. Shao, P. Wu, X. Zhang, D. Dimitrov, and J.B. Liu, On the maximum ABC index of graphs with prescribed size and without pendent vertices, IEEE Access 6 (2018), 27604–27616, .

• Crossref
• Export Citation
• [25]

S. Wang, Z. Shao, J.B. Liu, and B. Wei, The bounds of vertex Padmakar-Ivan index on k-trees, Mathematics 7 (2019), no. 4, 324, .

• Crossref
• Export Citation
• [26]

E.T. Boskoro, A.A.G. Ngurah, and R. Simanjantuk, On super (a, d)-EMT labeling of subdivision of K 1,3, SUT J. Math. 43 (2007), 127–136, .

• Crossref
• Export Citation
• [27]

Z. Shao, M. Liang, and X. Xu, Some new optimal generalized Sidon sequences, Ars Combin. 107 (2012), 369–378, .

• Crossref
• Export Citation
• [28]

J.C. Bermond, M. Cosnard, and S. Perennes, Directed acyclic graphs with the unique dipath property, Theor. Comput. Sci. 504 (2013), 5–11, .

• Crossref
• Export Citation
• [29]

Z. Shao, J. Amjadi, S.M. Sheikholeslami, and M. Valinavaz, On the total double Roman domination, IEEE Access 7 (2019), 52035–52041, .

• Crossref
• Export Citation
• [30]

Z. Shao, P. Wu, Y. Gao, I. Gutman, and X. Zhang, On the maximum ABC index of graphs without pendent vertices, Appl. Math. Comput. 315 (2017), 298–312, .

• Crossref
• Export Citation
• [31]

B. Assaye, M. Alamneh, L.N. Mishra, and Y. Mebrat, Dual skew Heyting almost distributive lattices, Appl. Math. Nonlinear Sci. 4 (2019), no. 1, 151–162, .

• Crossref
• Export Citation
• [32]

S.M. Hosamani, B. Kulkarni, R.G. Boli, and V.M. Gadag, QSPR analysis of certain graph theocratical matrices and their corresponding energy, Appl. Math. Nonlinear Sci. 2 (2017), no. 1, 131–150, .

• Crossref
• Export Citation
• [33]

S. Aidara and Y. Sagna, BSDEs driven by two mutually independent fractional Brownian motions with stochastic Lipschitz coefficients, Appl. Math. Nonlinear Sci. 4 (2019), no. 1, 139–150, .

• Crossref
• Export Citation

If the inline PDF is not rendering correctly, you can download the PDF file here.

• [1]

E.G. Goodaire and D.A. Robinson, Some special conjugacy closed loops, Canad. Math. Bull. 33 (1990), no. 1, 73–78, .

• Crossref
• Export Citation
• [2]

M.J. Osborn, Loops with the weak inverse property, Pacific J. Math. 10 (1960), no. 1, 295–304, .

• Crossref
• Export Citation
• [3]

R. Artzy, Inverse-cycles in weak-inverse loops, Proc. Amer. Math. Soc. 68 (1978), no. 2, 132–134, .

• Crossref
• Export Citation
• [4]

A. Drapal, Conjugacy closed loops and their multiplication groups, J. Algebra 272 (2004), no. 2, 838–850, .

• Crossref
• Export Citation
• [5]

E.L. Wilson, A class of loops with the isotopy-isomorphy property, Canad. J. Math. 18 (1966), 589–592, .

• Crossref
• Export Citation
• [6]

A. Grishkov and V.A. Zavarnitsine, Lagrange’s theorem for Moufang loops, Math. Proc. Cambridge Philos. Soc. 139 (2005), no. 1, 41–57, .

• Crossref
• Export Citation
• [7]

P.G. Nagy and P. Vojtěchovský, Automorphism groups of simple Moufang loops over perfect fields, Math. Proc. Cambridge Philos. Soc. 135 (2003), no. 2, 193–197, .

• Crossref
• Export Citation
• [8]

R.J. Shajila and S. Vimala, Graceful labelling for complete bipartite fuzzy graphs, Br. J. Math. Comput. Sci. 22 (2017), no. 2, Article no. BJMCS.32242, .

• Crossref
• Export Citation
• [9]

J.E. Hopcroft and R.M. Karp, An n 5/2 algorithm for maximum matchings in bipartite graphs, SIAM J. Comput. 2 (1973), 225–231, .

• Crossref
• Export Citation
• [10]

S. Fields and O. Song, A novel genetic system to detect protein-protein interactions, Nature 340 (1989), 245–246, .

• Crossref
• Export Citation
• [11]

A. Davy, P. Bello, N. Thierry-Mieg, P. Vaglio, J. Hitti, L. Doucette-Stamm, et al., A protein-protein interaction map of the Caenorhabditis elegans $26S$ proteasome, EMBO Rep. 2 (2001), no. 9, 821–828, .

• Crossref
• Export Citation
• [12]

T. Ito, T. Chiba, R. Ozawa, M. Yoshida, M. Hattori, and Y. Sakaki, A comprehensive two-hybrid analysis to explore the yeast protein interactome, Proc. Natl. Acad. Sci. USA 98 (2001), no. 8, 4569–4574, .

• Crossref
• Export Citation
• [13]

S. McCraith, T. Holtzman, B. Moss, and S. Fields, Genome-wide analysis of vaccinia virus protein-protein interactions, Proc. Natl. Acad. Sci. USA 97 (2000), no. 9, 4879–4884, .

• Crossref
• Export Citation
• [14]

J.C. Rain, L. Selig, H. de Reuse, V. Battaglia, C. Reverdy, S. Simon, et al., The protein-protein interaction map of Helicobacter pylori, Nature 409 (2001), 211–215, .

• Crossref
• Export Citation
• [15]

X. Shi, M.K. Hassanzadeh-Aghdam, and R. Ansari, Viscoelastic analysis of silica nanoparticle-polymer nanocomposites, Composites Part B: Eng. 158 (2019), 169–178, .

• Crossref
• Export Citation
• [16]

Z. Wang, M. Gerstein, and M. Snyder, RNA-Seq: a revolutionary tool for transcriptomics, Nat. Rev. Genet. 10 (2009), no. 1, 57–63, .

• Crossref
• Export Citation
• [17]

D. Yu, M. Kim, G. Xiao, and T.H. Hwang, Review of biological network data and its applications, Genomics Inform. 11 (2013), no. 4, 200–210, .

• Crossref
• Export Citation
• [18]

O. Sinanoglu, Theory of chemical reaction networks. All possible mechanisms or synthetic pathways with given number of reaction steps or species, J. Am. Chem. Soc. 97 (1975), no. 9, 2309–2320, .

• Crossref
• Export Citation
• [19]

J.L. Guillaume and M. Latapy, Bipartite graphs as models of complex networks, Phys. A 371 (2006), no. 2, 795–813, .

• Crossref
• Export Citation
• [20]

M.S. Sadi, P.K. Bachar, P. Ghosh, and M.S. Rahman, A new error correction coding approach, J. Adv. Inf. Technol. 4 (2013), no. 3, 142–147, .

• Crossref
• Export Citation
• [21]

A. Betten, G. Brinkmann, and T. Pisanski, Counting symmetric configurations V 3, Discrete Appl. Math. 99 (2000), no. 1-3, 331–338, .

• Crossref
• Export Citation
• [22]

Z. Shao, M.K. Siddiqui, and M.H. Muhammad, Computing Zagreb indices and Zagreb polynomials for symmetrical nanotubes, Symmetry 10 (2018), no. 7, 244, .

• Crossref
• Export Citation
• [23]

J.A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. #DS6 (2010), 246 pages, Available: http://www.combinatorics.org.

• [24]

Z. Shao, P. Wu, X. Zhang, D. Dimitrov, and J.B. Liu, On the maximum ABC index of graphs with prescribed size and without pendent vertices, IEEE Access 6 (2018), 27604–27616, .

• Crossref
• Export Citation
• [25]

S. Wang, Z. Shao, J.B. Liu, and B. Wei, The bounds of vertex Padmakar-Ivan index on k-trees, Mathematics 7 (2019), no. 4, 324, .

• Crossref
• Export Citation
• [26]

E.T. Boskoro, A.A.G. Ngurah, and R. Simanjantuk, On super (a, d)-EMT labeling of subdivision of K 1,3, SUT J. Math. 43 (2007), 127–136, .

• Crossref
• Export Citation
• [27]

Z. Shao, M. Liang, and X. Xu, Some new optimal generalized Sidon sequences, Ars Combin. 107 (2012), 369–378, .

• Crossref
• Export Citation
• [28]

J.C. Bermond, M. Cosnard, and S. Perennes, Directed acyclic graphs with the unique dipath property, Theor. Comput. Sci. 504 (2013), 5–11, .

• Crossref
• Export Citation
• [29]

Z. Shao, J. Amjadi, S.M. Sheikholeslami, and M. Valinavaz, On the total double Roman domination, IEEE Access 7 (2019), 52035–52041, .

• Crossref
• Export Citation
• [30]

Z. Shao, P. Wu, Y. Gao, I. Gutman, and X. Zhang, On the maximum ABC index of graphs without pendent vertices, Appl. Math. Comput. 315 (2017), 298–312, .

• Crossref
• Export Citation
• [31]

B. Assaye, M. Alamneh, L.N. Mishra, and Y. Mebrat, Dual skew Heyting almost distributive lattices, Appl. Math. Nonlinear Sci. 4 (2019), no. 1, 151–162, .

• Crossref
• Export Citation
• [32]

S.M. Hosamani, B. Kulkarni, R.G. Boli, and V.M. Gadag, QSPR analysis of certain graph theocratical matrices and their corresponding energy, Appl. Math. Nonlinear Sci. 2 (2017), no. 1, 131–150, .

• Crossref
• Export Citation
• [33]

S. Aidara and Y. Sagna, BSDEs driven by two mutually independent fractional Brownian motions with stochastic Lipschitz coefficients, Appl. Math. Nonlinear Sci. 4 (2019), no. 1, 139–150, .

• Crossref
• Export Citation
OPEN ACCESS

### Open Mathematics

Open Mathematics is a fully peer-reviewed, open access, electronic journal that publishes significant and original works in all areas of mathematics. The journal publishes both research papers and comprehensive and timely survey articles. Open Math aims at presenting high-impact and relevant research on topics across the full span of mathematics.