1.1 Introduction
Chapter one of these studies comprises of five major sub-sections which are research questions, the background of the research problem, research problem to be solved, and objectives to be satisfied by the research, the scope and significance of the study which emphasizes why the research topic is essential.
1.2 Background of the problem
Group theory is the study of groups that involves systems of the set of elements and binary operation applicable to two elements of sets that satisfy specific axioms. Geometric group theory describes a graph of groups as objects comprising of a collection of groups indexed by the edges and the vertices of a graph composed of a family of monomorphisms of the edge groups into the vertex groups. A fundamental group is a very unique group canonically linked to every finite connected graph of groups. Letting G be a fundamental group of a graph of groups with underlying graph Y . If G = π(Y , T). Then there exists a tree X, on which G acts without inversion on edges such that the factor graph G\X is isomorphic to Y and stabilizers of the vertices and edges in X are conjugate to the canonical images in G of the groups Gv and σ(Ge ) (Azad et al. 2016). From the analysis of different graph-related studies, graphs are key when analyzing fundamental groups and different components or features related to the graph of groups (Duan & Su 2012, January). Also, the linkage between graphs and groups has proven to be essential in conducting different studies as far as algebraic graph theory is concerned. Among the features brought by the linkage between graphs and groups are the minimum and maximum energies of graphs. There existing numerous types of graphs in graph theory which vary depending on their specific features, the number of edges, vertices, colours, their overall structures, and their interconnectivity.
1.2.1 Bipartite graphs
Bipartite graphs are also known as a bigraph is a type of graph whose vertices can be divided into two disjoint and independent sets U and V which are also regarded as parts of the graph such that every edge connects a vertex in V to U. Similarly, bigraphs are also described as the graphs without any odd-length cycles (Asratian et al. 2018). The two sets V and U can be described using two colours, such that when a single colour all nodes in U is blue and all nodes in V are green. The edges contain endpoints with different colours which are not there in a non-bipartite graph like a triangle which will contain a different colour in the third vertex. G=(U, V, E) is often written to denote a bipartite graph whose partition has the parts U and V with E denoting the edges of the graph. A non-connected bipartite graph contains multiple bipartitions in this case (U, V, E) is essential in specifying the most applicable bipartitions. In addition, a balanced bipartite graph (G) tends to exist when two subsets have equal cardinality |U| = |V| and is referred to as a biregular when all the vertices on the same side of the bipartition have the same degree. The following graph can therefore be said to be a good example of a bipartite graph because its vertices can be decomposed into two different sets X = {A, C} and Y = {B, D}. Also, the vertices of set Y join only with the vertices of set X and vice versa while the vertices within the same set do not join (Pavlopoulos et al. 2018).
Figure 1: (a,b,c)-Examples of closed bipartite graphs
1.2.2 Degree
Denoted as deg (v), the degree energy of a graph is referred to as the number of adjacent vertices for a vertex.
The degree sequence of a bipartite is the pair of the list each having two parts U and V. An example a complete bipartite graph K 2, 5 has a degree sequence ( 5, 5), (2, 2, 2, 2, 2). Isomorphic bipartite graphs have the same degree sequence; however, in some cases, non-isomorphic bipartite graphs may have the same degree sequence (Frieze & Melsted 2012).
1.3 Problem statement
A lot of bipartite graphs have been previously conducted most analysing different types of bipartite graphs and how they can be arrived at, and some focusing on how to find the minimum and maximum degree energy of some specific bipartite graphs. However, there is a very huge gap as far as finding the degree energy of commonly used graphs in association to common graphical groups. Bipartite graphs, being among the graphs having the most commonly used properties, with a wide range of unexplored applications makes a better choice about the fundamental group of graphs. Another common problem, specifically associated with the bipartite graphs is the bipartite realization problem which emphasises the difficulty experienced when looking for a simple bipartite graph with a maximum degree sequence or a degree sequence being two when provided with a list of natural numbers. To fill the research gap this study will use both quantitative and qualitative research approaches to analyse the maximum degree of energy in the fundamental group of bipartite graphs and determine some of their common applications in solving real-world problems.
1.4 Research questions
- What are the different types and features of bipartite graphs?
- What is the maximum degree of energy of the fundamental group of bipartite graphs?
- What is the application of maximum degree energy of the fundamental group of bipartite graphs?
1.5 Research Objectives
The objectives of this study are as follows:
- To understand the graph theory and different types of graphs of groups.
- To understand different types and components of bipartite graphs.
- To determine the maximum degree of energy of bipartite graphs.
- To determine different applications of maximum degree energy of the maximum degree of energy of bipartite graphs.
1.6 Scope of the study
This research is based on the analysis of different other previously related work about bipartite graphs and finding their absolute values as well as looking into different graphical theories to determine the maximum degree of energy of the fundamental group of bipartite graphs and their applications in different real-life situations. The research is conducted for a period of 6 months in which different previously research articles in relate graphical topics will be analysed and a comparison with the existing proved theories made and used to answer the research questions focused on in the study.
1.7 Significance of the study
In this study, different properties of bipartite graphs their features and the maximum degree energy of the fundamental group of bipartite graphs will be determined. The study will then analyse how the determined maximum degree energy of the respective bipartite graphs can offer solutions to real-life problems.
CHAPTER 2: LITERATURE REVIEW
2.1 Introduction
The research on different features of bipartite graphs have been previously done by other researchers. In this chapter a deeper look into some of the work specifically focusing on the properties of bipartite graphs, their types, and other specific properties like degree energies and their functional groups.
2.2 Bipartite graphs
According to Jimmy Salvatore (2007) each graph has specific properties that are used to describe them, however, something that seems to be common and applicable to all the types of and frequently used in graph theory is the degree of a vertex
(Jimmy Salvatore, 2007). The author further emphasises that the degree of a vertex, represented by deg(ῡ) = r where r represents the number of vertices adjacent to ῡ, in a graph represent the number of edges incident to it or the number of vertices adjacent to it. This argument is supported by Xiangnan et al 2016, who emphasises the importance of graphs to solving real-world problems and went further describing bipartite graphs as a graph that does not allow any edge to connect to vertices of the same set or a graph whose vertices can be broken down into two independent sets. U and V such that a specific edge (u,v) connects the vertex from V to U or from U to V.
Figure 2: vertex and edge connection in bipartite graphs
Serratosa (2014) describes bipartite graphs/bi-graphs as a set of graph vertices that acts as a meeting point for multiple lines which are disintegrated into two separate sets (Serratosa, 2014). Since the sets share nothing in common the two graph vertices falling on the set are therefore adjacent to each other or are connected by an edge. Serratosa 2014, further clarifies that bipartite graphs are equivalent to two colourable graphs or a special case of the k-partite graph with k=2.
Figure 3: sets of bipartite graphs
Heggernes et al. 2013 further characterise bipartite graphs using the following major considerations:
- A graph can be described to be bipartite if and only if it lacks an odd cycle.
- A graph is considered bipartite if and only if its chromatic number is equal or less than two or it is 2-colourable
- The only graphs with the symmetric spectrum are bipartite
Archdeacon et al 2010 emphasises that there are two major categories of bipartite graphs, cyclic and bicyclic bipartite graphs. Some common examples of these categories of graphs are gear, grid, and star graphs.
Figure 4: (a, b, c): Examples of Bipartite graphs
2.3 Degree energy of bipartite graphs
According to graph theory, the valency or the degree of a graph is the total number of edges incident to the vertex. Hosamani & Ramane 2016 further clarifies that the maximum degree energy of a graph is the degree of the vertex with the greatest number of edges incident to it. Adiga & Smitha 2009 uses different graphical theorem to analyse the maximum degree energy of some classes of graphs including a wider look into a bipartite graph with two sets V1 and V2. He further clarifies that if the set V1 comes first then the maximum degree matrix of the bipartite graph will take the form below:
Adiga & Smitha 2009 further clarifies that when given a bipartite graph G with eigenvalue µ concerning the maximum degree matrix with the multiplicity m then -µ is also an eigenvalue with multiplicity m.
CHAPTER 3: RESEARCH METHODOLOGY
3.1 Introduction
This chapter focuses on analysing the methodology that will be used to complete the research. The main aim of this research is to determine the maximum degree energy in the fundamental group of bipartite graphs as well as its application to solving real world problems. To facilitate this, this chapter is presented in three major parts, the research design, strategy and schedule using charts and tables. The research design introduces different phases and procedures followed in the research, the second and third parts provide the framework and schedule for each phase.
3.2 Research Design and procedure
| PHASES | PROCEEDURE |
| Phase 1 | Gap identification: –
· Literature review on different types of graphs, their unique properties and the level of research previous researchers have done on each of the identified graphs. Ø Formulation of problem statement and objectives on the identified gap.
|
| Phase 2 | Literature review on concepts in bipartite graphs, different types and their features.
Ø To understand different unique features of bipartite graphs. Ø To determine different categories of bipartite graphs and how applicable they are in real life scenarios. Ø To determine the generalized connectivity of the line graph and the total graph for the complete bipartite graph. Ø To understand the appropriate labelling of edges and vertices f bipartite graphs. Ø To understand how maximum degree energy is calculated and how essential it is in real life scenario. |
| Phase 3 | Compute absolute degree energies of different types of bipartite graphs.
· To determine the maximum degree energy of each type or category of bipartite graphs. |
| Phase 4 | Relating the maximum degree energies obtained with different real life applications and recommend the most appropriate application. |
Figure 5: Research design
3.2 Research Strategies
Figure 6: Work plan
3.3 Project time-line
Figure 7: Project time line
CHAPTER 4: STATUS OF THE RESEARCH
4.1 Expected Findings
Based on the objectives and research problems that ought to be solved by this study the following are expected:
- A clear understanding and description of the different types and features of bipartite graphs.
- Fundamental groups of bipartite graphs
- The maximum degree energy of the fundamental group of bipartite graphs.
- Real-world application of maximum degree energy of the fundamental group of bipartite graphs,
4.2 Progress.
Currently, the first two phases of the research are done where preliminary research assessment and gap identification has been done as well as formulation of the research problem and objectives and the most applicable research plan and methodology that will lead to the most reliable solution drafted. Also, some part of the literature review has so far been done, this includes a deeper look into bipartite graphs, their features and other relevant components and the fundamental groups of bipartite graphs.
