This book is intended as an introduction to graph theory. Our aim has Many people have contributed, either directly or indirectly, to this book. The notes form the base text for the course ”MAT Graph Theory”. of course many modern text-books with similar contents, e.g. the popular GROSS. an introductory graph theory course aiming to equip its audience for the most likely developments ahead? I have tried in this book to offer material for such a.

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A catalogue record for this book is available from the British Library In recent years, graph theory has established itself as an important mathematical tool in. Introduction to Graph Theory, by Douglas B. West. A few solutions have .. of the Petersen graph on the cover of the book. There are many. contained in this book and cannot accept any legal responsibility or The links between graph theory and other branches of mathematics are.

THEOREM In a connected graph G with exactly 2k odd vertices, there exist k edge-disjoint subgraphs such that they together contain all edges of G and that each is a unicursal graph.

Let the odd vertices of the given graph G be named v1, v2, …, vk; w1, w2, …, wk in any arbitrary order. Add k edges to G between the vertex pairs v1, w1 , v2, w2 , …, vk, wk to form a new graph G'. Since every vertex of G' is of even degree, G' consists of an Euler line p.

Now if we remove from p the k edges we just added no two of these edges are incident on the same vertex , p will be split into k walks, each of which is a unicursal line: The first removal will leave a single unicursal line; the second removal will split that into two unicursal lines; and each successive removal will split a unicursal line into two unicursal lines, until there are k of them.

Thus the theorem. Removal of any one edge from a Hamiltonian circuit generates a path. This path is called Hamiltonian path. The subgraph of a complete graph of n vertices in Figure is a Hamiltonian circuit. Observe that each rotation produces a Hamiltonian circuit that has no edge in Trees with 1, 2, 3, and 4 vertices are shown in figure.

Decision tree shown in figure There is one and only one path between every pair of vertices in a tree T.

In a graph G there is one and only one path between every pair of vertices, G is a tree. A tree with n vertices has n-1 edges. Any connected graph with n vertices has n-1 edges is a tree. A graph is a tree if and only if it is minimally connected.

A graph G with n vertices has n-1 edges and no circuits are connected. Since T is a connected graph, there must exist at least one path between every pair of vertices in T. Now suppose that between two vertices a and b of T there are two distinct paths.

The union of these two paths will contain a circuit and T cannot be a tree. Existence of a path between every pair of vertices assures that G is connected. A circuit in a graph with two or more vertices implies that there is at least one pair of vertices a, b such that there are two distinct paths between a and b.

Since G has one and only one path between every pair of vertices, G can have no circuit. Therefore, G is a tree. Prepared by G.

Appasami, Assistant professor, Dr. The theorem will be proved by induction on the number of vertices. Assume that the theorem holds for all trees with fewer than n vertices. In T let ek be an edge with end vertices vi and vj. According to Theorem , there is no other path between vi and vj, except ek. Therefore, deletion of ek from T will disconnect the graph, as shown in Figure.

Furthermore, T — ek consists of exactly two components, and since there were no circuits in no begin with, each of these components is a tree. Both these trees, t1 and t2, have fewer than n vertices each, and therefore, by the induction hypothesis, each contains one less edge than the number of vertices in it. Thus T — ek consists of n — 2 edges and n vertices. Hence T has exactly n — 1 edges.

The shortest paths between vertices v6 and v2 are a, e and b, f , each of length two. Finding Center of graph. The function f x, y of two variables defines the distance between them. These function must satisfy certain requirements. They are 1. Triangle inequality: Radius and Diameter in a tree. The eccentricity of a center in a tree is defined as the radius of tree.

The length of the longest path in a tree is called the diameter of tree. In general tree means without any root. They are sometimes called as free trees non rooted trees. The root is enclosed in a small triangle.

All rooted trees with four vertices are shown below. Rooted binary tree There is exactly one vertex of degree two root and each of remaining vertex of degree one or three. A binary rooted tree is special kind of rooted tree.

Thus every binary tree is a rooted tree. A non pendent vertex in a tree is called an internal vertex. A tree T is said to be a spanning tree of a connected graph G if T is a subgraph of G and T contains all vertices maximal tree subgraph.

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Branch and chord. An edge in a spanning tree T is called a branch of T. An edge of G is not in a given spanning tree T is called a chord tie or link. Edge e1 is a branch of T Edge e5 is a chord of T 2. Complement of tree. Rank and Nullity: Spanning Tree T: Complement of Tree T v3 v1 v2 v4 e2 e3 e4 e5 e6 e1 e7 v3 v1 v2 v4 e3 e4 e1 Graph G: This is because there already exists a path between any two vertices of a tree. If the branches of the spanning tree T of a connected graph G are b1,.

We call this circuit a fundamental circuit. Every fundamental circuit has exactly one link which is not in any other fundamental circuit in the fundamental set of circuits. Therefore, we can not write any fundamental circuit as a ring sum of other fundamental circuits in the same set.

In other words, the fundamental set of circuits is linearly independent under the ring sum operation. The corresponding vertex sets are V1 and V2. Then, v1,v2 is a cut of G.

Thus, every branch bi of T has a corresponding cut set Ii. The cut sets I1,. Every fundamental cut set includes exactly one branch of T and every branch of T belongs to exactly one fundamental cut set. Therefore, every spanning tree defines a unique fundamental set of cut sets for G. Continuing from the previous example. The graph has the spanning tree that defines these fundamental cut sets: Counter hypothesis: Some cut set F of G does not include any branches of a spanning tree T.

If G is a weighted graph, then the weight of a spanning tree T of G is defined as the sum of the weights of all the branches in T. A spanning tree with the smallest weight in a weighted graph is called a shortest spanning tree shortest-distance spanning tree or minimal spanning tree.

A spanning arborescence in a connected digraph is a spanning tree that is an arborescence. Therefore D1 is a spanning arborescence in D and is rooted at v. A minimal set of edges in a connected graph whose removal reduces the rank by one is called minimal cut set simple cut-set or cocycle.

Every edge of a tree is a cut set. Properties of circuits and cut sets: Every cut set of a connected graph G includes at least one branch from every spanning tree of G. The edge set F of the connected graph G is a cut set of G if and only if i F includes at least one branch from every spanning tree of G, and e d cb v1 v3 v2 v5 v4 g a h Graph G: Let us first consider the case where F is a cut set.

Then, i is true previous proposition a. This T is also a spanning tree of G. Hence, ii is true. Let us next consider the case where both i and ii are true. Hence, F is a cut set.

For this purpose we list all cut-sets of the corresponding graph, and find which ones have the smallest number of edges. It must also have become apparent to you that even in a simple example, such as in Fig. There is a large number of cut-sets, and we must have a systematic method of generating all relevant cut-sets.

In the case of circuits, we solved a similar problem by the simple technique of finding a set of fundamental circuits and then realizing that other circuits in a graph are just combinations of two or more fundamental circuits.

We shall follow a similar strategy here. Just as a spanning tree is essential for defining a set of fundamental circuits, so is a spanning tree essential for a set of fundamental cut-sets. It will be beneficial for the reader to look for the parallelism between circuits and cut-sets. Fundamental Cut-Sets: Consider a spanning tree T of a connected graph G. Take any branch b in T. Consider the same partition of vertices in G, and the cut set S in G that corresponds to this partition.

Cut-set S will contain only one branch b of T, and the rest if any of the edges in S are chords with respect to T. Such a cut-set S containing exactly one branch of a tree T is called a fundamental cut-set with respect to T. Sometimes a fundamental cut-set is also called a basic cut-set.

Just as every chord of a spanning tree defines a unique fundamental cir-cuit, every branch of a spanning tree defines a unique fundamental cut-set. It must also be kept in mind that the term fundamental cut-set like the term fundamental circuit has meaning only with respect to a given spanning tree. Now we shall show how other cut-sets of a graph can be obtained from a given set of cut-sets. There is a distinct fundamental cycle for each edge; thus, there is a one-to-one correspondence between fundamental cycles and edges not in the spanning tree.


Dual to the notion of a fundamental cycle is the notion of a fundamental cutset. By deleting just one edge of the spanning tree, the vertices are partitioned into two disjoint sets. The fundamental cutset is defined as the set of edges that must be removed from the graph G to accomplish the same partition. Let c, be a chord with respect to T, and let the fundamental circuit made by ei be called r, con-sisting of k branches b1, b2,.

Every branch of any spanning tree has a fundamental cut-set associated with it.

Because of above Theorem , there must be an even number of edges common to and Si. Edge b, is in both and Si and there is only one other edge in which is c, that can possibly also be in Si. Therefore, we must have two edges b, and c, common to S, and r. Thus the chord c, is one of the chords c1, , C4.

Exactly the same argument holds for fundamental cut-sets associated with b2,b3,. Therefore, the chord c, is contained in every fundamental cut-set associated with branches in.

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Is it possible for the chord c, to be in any other fundamental cut-set S' with respect to T, of course besides those associated with b2,b3,. The answer is no. Otherwise since none of the branches in r are in S' , there would be only one edge c, common to S' and , a contradiction to Theorem.

Thus we have an important result. THEOREM With respect to a given spanning tree T, a chord ci that determines a fundamental circuit occurs in every fundamental cut-set associated with the branches in and in no other.

The three fundamental cut-sets determined by the three branches e, h, and k are determined by branch e: Chord f occurs in each of these three fundamental cut-sets, and there is no other fundamental cut-set that contains f. The converse of above Theorem is also true.

Each cut-set of a connected graph G consists of certain number of edges. The number of edges in the smallest cut-set is defined as the edge Connectivity of G. The edge Connectivity of a connected graph G is defined as the minimum number of edges whose removal reduces the rank of graph by one. The edge Connectivity of a tree is one.

The edge Connectivity of the above graph G is three. The vertex Connectivity of a tree is one. The vertex Connectivity of the above graph G is one. A connected graph is said to be separable graph if its vertex connectivity is one.

All other connected graphs are called non-separable graph. Non-Separable Graph H: A separable graph consists of two or more non separable subgraphs. Each of the largest nonseparable is called a block or component. The above graph has 5 blocks. If a connected graph G has no separating sets, then it is a complete graph.

Clearly, if G is connected, then it is 1-connected. The amount of flow on an edge cannot exceed the capacity of the edge. The max. Various transportation networks or water pipelines are conveniently represented by weighted directed graphs.

These networks usually possess also some additional requirements.

Basic Graph Theory

Goods are transported from specific places warehouses to final locations marketing places through a network of roads. In modeling a transportation network by a digraph, we must make sure that the number of goods remains the same at each crossing of the roads.

The problem setting for such networks was proposed by T. Harris in the s. According to this law, in every electrical network the amount of current flowing in a vertex equals the amount flowing out that vertex. We adopt the following notations: Two graphs G1 and G2 are said to be 1-Isomorphic if they become isomorphic to each other under repeated application of the following operation.

Operation 1: Graph G1: Graph G1 is 1-Isomorphism with Graph G2. Each of the largest non separable subgraphs is called a block.

Some authors use the term component, but to avoid confusion with components of a disconnected graph, we shall use the term block. The graph in Fig has two blocks.

The graph in Fig has five blocks and three cut-vertices a, b, and c ; each block is shown enclosed by a broken line. Note that a non separable connected graph consists of just one block. Visually compare the disconnected graph in Fig. These two graphs are certainly not isomorphic they do not have the same number of vertices , but they are related by the fact that the blocks of the graph in Fig.

Such graphs are said to be I-isomorphic. More formally: Two graphs G1 and G2 are said to be I-isomorphic if they become isomorphic to each other under repeated application of the following operation.

Operation I: From this Under operation 1, whenever a cut-vertex in a graph G is "split" into two vertices, the number of components in G increases by one. Therefore, the rank of G which is number of vertices in C - number of components in G remains invariant under operation 1.

Also, since no edges are destroyed or new edges created by operation I, two 1-isomorphic graphs have the same number of edges. What if we join two components of Fig by "gluing" together two vertices say vertex x to y? We obtain the graph shown in Fig. Clearly, the graph in Fig is 1-isomorphic to the graph in Fig.

Since the blocks of the graph in Fig are isomorphic to the blocks of the graph in Fig, these two graphs are also 1-isomorphic. Thus the three graphs in above Figs. Let vertices x1 and y1 go with g1 and vertices x2 and y2 go with g2.

Now rejoin the graphs g1 and g2 by merging x1 with y2 and x2 with y1. This combinatorial abstract object G can also be represented by means of a geometric figure. The figure is one such geometric representation of this graph G. Any graph can be geometrically represented by means of such configuration in three dimensional Euclidian space.

Drawing a graph in the plane without edge crossing is called embedding the graph in the plane or planar embedding or planar representation. Given a planar representation of a graph G, a face also called a region is a maximal section of the plane in which any two points can be joint by a curve that does not intersect any part of G. When we trace around the boundary of a face in G, we encounter a sequence of vertices and edges, finally returning to our final position.

Let v1, e1, v2, e2,. Some edges may be encountered twice because both sides of them are on the same face. A tree is an extreme example of this: It has been indicated that a graph can be represented by more than one geometrical drawing. In some drawing representing graphs the edges intersect cross over at points which are not vertices of the graph and in some others the edges meet only at the vertices.

On the other hand, a graph which cannot be represented by a plane drawing in which the edges meet only at the vertices is called a non planar graph. In other words, a non planar graph is a graph whose every possible plane drawing contains at least two edges which intersect each other at points other than vertices.

Example 1 Show that a a graph of order 5 and size 8, and b a graph of order 6 and size 12, are planar graphs. A graph of order 5 and size 8 can be represented by a plane drawing. Graph a Graph b Similarly, fig. It has been two hundred and eighty years till now. Fractional Graph Theory Edward R. Scheinerman and Daniel H. Ullman PDF Pages English Graph theory is one of the branches of modern mathematics having experienced a most impressive development in recent years.

This book will draw the attention of the combinatorialists to a wealth of new problems and conjectures. Topics covered includes: General Theory: Graph Theory Notes Dr. Graph Theory by Keijo Ruohonen Keijo Ruohonen PDF Pages English This note contains an introduction to basic concepts and results in graph theory, with a special emphasis put on the network-theoretic circuit-cut dualism.

Graph Theory with Applications J. Bondy and U. Murty Online NA Pages English The primary aim of this book is to present a coherent introduction to graph theory, suitable as a textbook for advanced undergraduate and beginning graduate students in mathematics and computer science. This note covers the following topics: Chung Fan R. Eigenvalues and the Laplacian of a graph, Isoperimetric problems, Diameters and eigenvalues, Eigenvalues and quasi-randomness.

Definitions for graphs, Exponential generating functions, egfs for labelled graphs, Unlabelled graphs with n nodes and Probability of connectivity 1. Free Graph Theory Books. Abstract Algebra. Linear Algebra.

Algebraic Graph Theory

Commutative Algebra. Complex Algebra. Elliptic Curves. Geometric Algebra. Groups Theory. Higher Algebra. Graph Colorings. Triangulated Graphs.

Back Matter Pages About this book Introduction Graph theory has experienced a tremendous growth during the 20th century. One of the main reasons for this phenomenon is the applicability of graph theory in other disciplines such as physics, chemistry, psychology, sociology, and theoretical computer science. This book aims to provide a solid background in the basic topics of graph theory.

It covers Dirac's theorem on k-connected graphs, Harary-Nashwilliam's theorem on the hamiltonicity of line graphs, Toida-McKee's characterization of Eulerian graphs, the Tutte matrix of a graph, Fournier's proof of Kuratowski's theorem on planar graphs, the proof of the nonhamiltonicity of the Tutte graph on 46 vertices and a concrete application of triangulated graphs. The book does not presuppose deep knowledge of any branch of mathematics, but requires only the basics of mathematics.

It can be used in an advanced undergraduate course or a beginning graduate course in graph theory.This path is called Hamiltonian path. Geometric Algebra. The eccentricity of a center in a tree is defined as the radius of tree. Immersion and embedding of 2-regular digraphs, Flows in bidirected graphs, Average degree of graph powers, Classical graph properties and graph parameters and their definability in SOL, Algebraic and model-theoretic methods in constraint satisfaction, Coloring random and planted graphs: Goods are transported from specific places warehouses to final locations marketing places through a network of roads.

Graph Theory with Applications J. Harmonic Analysis. For example, the following two graphs are edge-disjoint sub-graphs of the graph G.