Introduction to graph theory (All chapter)
- 2. Graphs serve as mathematical models to analyze many concrete real-world problems successfully. Certain problems in physics, chemistry, communication science, computer technology, genetics, psychology, sociology, and linguistics can be formulated as problems in graph theory. Also, many branches of mathematics, such as group theory, matrix theory, probability, and topology, have close connections with graph theory. Some puzzles and several problems of a practical nature have been instrumental in the development of various topics in graph theory. The famous Königsberg bridge problem has been the inspiration for the development of Eulerian graph theory. The challenging Hamiltonian graph theory has been developed from the “Around the World” game of Sir William Hamilton. The theory of acyclic graphs has been developed for solving problems of electrical networks, and the study of “trees” was developed for enumerating isomers of organic compounds.
- 3. The well-known four-color problem formed the very basis for the development of planarity in graph theory and combinatorial topology. Problems of linear programming and operations research (such as maritime traffic problems) can be tackled by the theory of flows in networks. Kirkman’s schoolgirl problem and scheduling problems are examples of problems that can be solved by graph colorings. The study of simplicial complexes can be associated with the study of graph theory. Many more such problems can be added to this list.
- 4. Directed graphs arise in a natural way in many applications of graph theory. The street map of a city, an abstract representation of computer programs, and network flows can be represented only by directed graphs rather than by graphs. Directed graphs are also used in the study of sequential machines and system analysis in control theory.
- 5. The connectivity of a graph is a “measure” of its connectedness. Some connected graphs are connected rather “loosely” in the sense that the deletion of a vertex or an edge from the graph destroys the connectedness of the graph. There are graphs at the other extreme as well, such as the complete graphs Kn , n ≥ 2, which remain connected after the removal of any k–vertices, 1 ≤ k ≤ n –1. Consider a communication network. Any such network can be represented by a graph in which the vertices correspond to communication centers and the edges represent communication channels. In the communication network of (Fig. 3.1a) any disruption in the communication center ѵ will result in a communication breakdown, whereas in the network of (Fig. 3.1b) at least two communication centers have to be disrupted to cause a breakdown. It is needless to stress the importance of maintaining reliable communication networks at all times, especially during times of war, and the reliability of a communication network has a direct bearing on its connectivity.
- 6. In this chapter, we study the two graph parameters, namely, vertex connectivity and edge connectivity. We also introduce the parameter cyclical edge connectivity. We prove Menger’s theorem and several of its variations. In addition, the theorem of Ford and Fulkerson on flows in networks is established.
- 7. “Trees” form an important class of graphs. Of late, their importance has grown considerably in view of their wide applicability in theoretical computer science. In this chapter, we present the basic structural properties of trees, their centers and centroids. In addition, we present two interesting consequences of the Tutte – Nash – Williams theorem on the existence of k–pairwise edge-disjoint spanning trees in a simple connected graph. We also present Cayley’s formula for the number of spanning trees in the labeled complete graph Kn. As applications, we present Kruskal’s algorithm and Prim’s algorithm, which determine a minimum-weights panning tree in a connected weighted graph and discuss Dijkstra’s algorithm, which determines a minimum–weight shortest path between two specified vertices of a connected weighted graph.
- 8. Vertex–independent sets and vertex coverings as also edge-independent sets and edge coverings of graphs occur very naturally in many practical situations and hence have several potential applications. In this chapter, we study the properties of these sets. In addition, we discuss matchings in graphs and, in particular, in bipartite graphs. Matchings in bipartite graphs have varied applications in operations research. We also present two celebrated theorems of graph theory, namely, Tutte’s 1-factor theorem and Hall’s matching theorem. All graphs considered in this chapter are loopless.
- 9. The study of Eulerian graphs was initiated in the 18th century and that of Hamiltonian graphs in the 19th century. These graphs possess rich structures: hence, their study is a very fertile field of research for graph theorists. In this chapter, we present several structure theorems for these graphs.
- 10. Graph theory would not be what it is today if there had been no coloring problems. In fact, a major portion of the 20th-century research in graph theory has its origin in the four-color problem. ( See Chap. 8 for details.) In this chapter, we present the basic results concerning vertex colorings and edge colorings of graphs. We present two important theorems on graph colorings, namely, Brooks’ theorem and Vizing’s theorem. We also present a brief discussion on “snarks” and Kirkman’s schoolgirl problem. In addition, a detailed description of the Mycielskian of a graph is also presented.
- 11. The study of planar and nonplanar graphs and, in particular, the several attempts to solve the four-color conjecture have contributed a great deal to the growth of graph theory. Actually, these efforts have been instrumental to the development of algebraic, topological, and computational techniques in graph theory. In this chapter, we present some of the basic results on planar graphs. In particular, the two important characterization theorems for planar graphs, namely, Wagner’s theorem (same as the Harary–Tutte theorem) and Kuratowski, s theorem, are presented. Moreover, the nonhamiltonicity of the Tutte graph on 46 vertices (see Fig. 8.28 and also the front wrapper) is explained in detail.
- 12. Triangulated graphs form an important class of graphs. They are a subclass of the class of perfect graphs and contain the class of interval graphs. They possess a wide range of applications. We describe later in this chapter an application of interval graphs in phasing the traffic lights at a road junction. We begin with the definition of perfect graphs.
- 13. “Domination in graphs” is an area of graph theory that has received a lot of attention in recent years. It is reasonable to believe that “domination in graphs” has its origin in “chessboard domination”. The “Queen Domination” problem asks : What is the minimum number of queens required to be placed on an (8 x 8) chessboard so that every square not occupied by any of these queens will be dominated (that is, can be attacked) by one of these queens ? Recall that a queen can move horizontally, vertically, and diagonally on the chessboard. The answer to the above question is 5. (Fig. 10.1) gives one set of dominating locations for the five queens.
- 14. In this chapter, we look at the properties of graphs from our knowledge of their eigenvalues. The set of eigenvalues of a graph G is known as the spectrum of G and denoted by Sp (G). We compute the spectra of some well-known families of graphs–the family of complete graphs, the family of cycles etc. We present Sachs’ theorem on the spectrum of the line graph of a regular graph. We also obtain the spectra of product graphs–Cartesian product, direct product, and strong product. We introduce Cayley graphs and Ramanujan graphs and highlight their importance. Finally, as an application of graph spectra to chemistry, we discuss the “energy of a graph”– a graph invariant that is widely studied these days. All graphs considered in this chapter are finite, undirected, and simple.