Graph Theory is a branch of discrete mathematics that studies graphs, which are mathematical structures used to model pairwise relations between objects. A graph consists of vertices (also called nodes) and edges (also called links) that connect pairs of vertices.
Key Concepts:
Vertices and Edges: The basic building blocks of a graph.
Types of Graphs: Includes directed, undirected, weighted, and unweighted graphs.
Paths and Cycles: Sequences of vertices where each pair is connected by an edge; cycles return to the starting vertex.
Connectivity: Describes how vertices are connected within a graph.
Graph Traversal: Algorithms like Depth-First Search (DFS) and Breadth-First Search (BFS) used to explore graphs.
Graph Coloring: Assigning colors to vertices so that no two adjacent vertices share the same color.
Planar Graphs: Graphs that can be drawn on a plane without edges crossing.
Applications: Graph theory is widely applied in computer science (e.g., networking, algorithms), biology (e.g., modeling neural networks), social sciences (e.g., analyzing social networks), and many other fields.
By studying graph theory, one can understand and solve complex problems related to network structures, optimization, and connectivity.
Course Description: This course covers the fundamentals of graph theory, a critical area in discrete mathematics with extensive applications in computer science, operations research, and network analysis. Students will learn about graph properties, various types of graphs, and algorithms used to solve graph-related problems. Emphasis will be placed on both theoretical understanding and practical applications.
Prerequisites:
Completion of an introductory Discrete Mathematics course
Basic understanding of mathematical proofs
Course Objectives:
Understand fundamental concepts and terminology of graph theory
Analyze and solve problems related to various types of graphs
Apply graph algorithms to practical problems in computer science and related fields
Develop skills in mathematical reasoning and proof techniques specific to graph theory
Part 1: Introduction to Graph Theory
Overview of graph theory and its applications
Basic definitions: graphs, vertices, edges
Types of graphs: simple, directed, undirected, weighted
Part 2: Graph Representations
Adjacency matrix
Adjacency list
Incidence matrix
Practical applications of graph representations
Part 3: Graph Traversals
Depth-first search (DFS)
Breadth-first search (BFS)
Applications of graph traversals
Part 4: Connectivity and Components
Connected and disconnected graphs
Strongly connected components
Weakly connected components
Applications in network analysis
Part 5: Eulerian and Hamiltonian Graphs
Eulerian paths and circuits
Hamiltonian paths and cycles
Algorithms to find Eulerian paths and Hamiltonian cycles
Part 6: Graph Coloring
Vertex coloring
Chromatic number
Applications in scheduling and register allocation
Part 7: Planar Graphs
Planarity and Kuratowski’s theorem
Euler’s formula for planar graphs
Graph drawing algorithms
Part 8: Trees and Spanning Trees
Properties of trees
Minimum spanning tree (MST)
Kruskal’s algorithm
Prim’s algorithm
Part 9: Network Flows
Maximum flow problem
Ford-Fulkerson algorithm
Applications in network routing and logistics
Part 10: Matchings and Coverings
Bipartite graphs
Maximum matching
Hall’s marriage theorem
Vertex cover and edge cover
Part 11: Advanced Topics in Graph Theory I - Graph isomorphism - Automorphisms - Cayley’s theorem
Part 12: Advanced Topics in Graph Theory II - Graph spectra and eigenvalues - Applications in chemistry and network analysis
Part 13: Algorithms and Complexity - Computational complexity of graph algorithms - NP-complete problems in graph theory - Approximation algorithms
Part 14: Applications of Graph Theory - Graph theory in computer networks - Social network analysis - Graph theory in biological networks
Graph theory is the study of points and lines. In particular, it involves the ways in which sets of points, called vertices, can be connected by lines, called edges.
Graphs are classified according to their complexity, the number of edges allowed between any two vertices, and whether or not directions are assigned to edges.
Graphs are classified according to their complexity, the number of edges allowed between any two vertices, and whether or not directions (for example, up or down) are assigned to edges.
Graph theory is the study of points and lines. In particular, it involves the ways in which sets of points, called vertices, can be connected by lines or arcs, called edges. Graphs in this context differ from the more familiar coordinate plots that portray mathematical relations and functions.
Graphs are classified according to their complexity, the number of edges allowed between any two vertices, and whether or not directions (for example, up or down) are assigned to edges. Various sets of rules result in specific properties that can be stated as theorems.
Graph theory has proven useful in the design of integrated circuits ( IC s) for computers and other electronic devices. These components, more often called chip s, contain complex, layered microcircuits that can be represented as sets of points interconnected by lines or arcs.
Using graph theory, engineers develop chips with maximum component density and minimum total interconnecting conductor length. This is important for optimizing processing speed and electrical efficiency.
Eulerian Paths and Cycles
Adjacency List: Using an adjacency list to construct a graph.
Adjacency List ( Worked Example)
An representation of a graph is a collection of unordered lists, one for each vertex in the graph. Each list describes the set of neighbors of its vertex.
An is a means of representing which vertices (or nodes) of a graph are adjacent to which other vertices. Another matrix representation for a graph is the incidence matrix.
The density of a graph G = (V,E) measures how many edges are in set E compared to the maximum possible number of edges between vertices in set V. Density is calculated as follows:
The average degree of a graph G is another measure of how many edges are in set E compared to number of vertices in set V. Because each edge is incident to two vertices and counts in the degree of both vertices, the average degree of an undirected graph is 2*|E|/|V|.
A directed graph is a graph where the arcs are one-directional, that is an arc between two nodes only indicates the possibility of moving from one node to the other, but not in the opposite direction. The arcs will usually be drawn as arrows to indicate the direction. An example of a potential use for a directed graph would be a graph which tracks possible states that a computer could be in; there may be a way for a computer in one state to reach a subsequent state, but no way to return from the second state to the first.
A weighted graph is a graph where there is some ‘cost’ associated with each arc. Usually, a little number will appear directly adjacent to every arc to express this price. A typical use of a weighted graph would be planning routes between locations on a map where the ‘cost’ associated with the arc would be the distance between the two locations.
A tree is a special graph which is connected (every node can be reached from every other node by following one or more arcs) and for which the number of arcs is exactly one fewer than the number of nodes. A tree is usually drawn with one node (called the ‘root node’) at the top of the diagram, and then ‘growing’ downwards with arcs and nodes getting further and further from the root. In this way, nodes can be grouped in terms of their distance from the root. Every node (aside from the root) is referred to as the ‘child’ of the node to which it is connected and which is one step closer to the root. This node is also called the ‘parent’ node of the child. Every node has at most one parent but can have any number of children. Nodes without any children are commonly called ‘leaf nodes’. A typical use of a tree would be a decision problem where the answer to a question determines the next question and set of possible answers.
A planar graph is a graph that can be drawn in the plane such that there are no edge crossings.
The planar graphs can be characterized by a theorem first proven by Kazimierz Kuratowski in 1930, which states that the planar graphs are exactly those graphs G such that \(K_5 \not \preceq G\) and \(K_{3,3</h5> \not \preceq G\) .
There are various types of graphs depending upon the number of vertices, number of edges, interconnectivity, and their overall structure. We will discuss only a certain few important types of graphs in this chapter.
Let \(G\) be a connected planar graph, and let V, E and R denote, respectively, the numbers of vertices, edges, and regions in a plane drawing of G. Then \(V - E + R = 2\).
In graph theory, the degree sequence of a graph is a list of the degrees of its vertices, usually written in non-increasing order. The degree of a vertex is the number of edges incident to it.
Essentially, the degree sequence provides a summary of the graph’s structure by indicating how many connections (or edges) each vertex (or node) has.
Consider a simple graph with the following vertices and edges: - Vertices: \(V = \{A, B, C, D, E\}\) - Edges: \(E = \{(A, B), (A, C), (B, D), (C, D), (D, E)\}\)
The degrees of the vertices are: - Vertex \(A\): 2 edges (connected to \(B\) and \(C\)) - Vertex \(B\): 2 edges (connected to \(A\) and \(D\)) - Vertex \(C\): 2 edges (connected to \(A\) and \(D\)) - Vertex \(D\): 3 edges (connected to \(B\), \(C\), and \(E\)) - Vertex \(E\): 1 edge (connected to \(D\))
The degree sequence is therefore: \[ (3, 2, 2, 2, 1) \]
In graph theory, a regular graph is a graph where each vertex has the same number of edges, meaning that every vertex has the same degree. If every vertex in the graph has degree \(k\), the graph is called a \(k\)-regular graph or simply a regular graph of degree \(k\).
Regular graphs are important in various fields, including network design and error-correcting codes, due to their uniform structure.
In graph theory, a planar graph is a graph that can be embedded in the plane, meaning it can be drawn on a flat surface without any of its edges crossing each other. Here are some key points about planar graphs:
Planar graphs are used in various fields such as network design, geography (map coloring), and computer graphics. They simplify problems involving circuit layout, urban planning, and geographical information systems.
Whether or not it is possible to traverse a graph from one vertex to another is dependent on how connected a graph is.
Theorem: Let G be a graph of order at least 3. If there are distinct vertices u and v in G so that G-u and G-v are both connected, then G is also connected. Proof: Let w be a G vertex, which is different from both u and v. We want to prove connectedness, i.e., that for every pair (x,y) of vertices in G, there is an x-y walk in G. We may consider three (partly overlapping) cases:
A graph G is called k-connected if for every \({\displaystyle S\subseteq V(G)}\), and ${|S|<k} $, \({\displaystyle G-S}\) is connected, and \({\displaystyle |G|>k}\).
Similarly, a graph G is called $ℓ {} $ edge-connected if for every \({\displaystyle S\subseteq E(G)}\) , and $ {|S|<} $ , G − S {G-S} is connected, and ${|G|>1} $ . The connectivity of G is the greatest k such that G is k-connected, and is denoted by ${(G)} $ . Relatedly, the edge-connectivity of G is the greatest ℓ {} such that G is ℓ {} edge-connected, and is denoted by \({\displaystyle \lambda (G)}\).
Theorem: Let G be a k-connected graph. Then, ∀ i ∈ N , 0 ≤ i ≤ k {i ,0ik} , G is i-connected. Proof: Since G is k-connected, for all \({\displaystyle S\subseteq V(G),|S|<k}\), \({\displaystyle G-S}\) is connected. Then, since \({\displaystyle i\leq k}\) , for all ${SV(G),|S|<ik} $ , G − S {G-S} is also connected.
Theorem: Let G be a nontrivial graph. Then, \({\displaystyle \lambda (G)\leq \delta (G)}\).
If G is $ {} $ edge-connected, show that G is also i edge-connected \({\displaystyle \forall i\in \mathbb {N} ,0\leq i\leq \ell }\) .
The two graphs have different numbers of vertices.
The two graphs have different numbers of edges.
One graph has parallel edges and the other does not.
One graph has a loop and the other does not.
One graph has a vertice of degree \(k\) (for example) and the other does not.
One graph is connected and the other is not.
One graph has a cycle and the other has not.
They have a different number of connected components
They have a different number of vertices
They have different degrees sequences
They have a different number of paths of any given length
They have a different number of cycles of any length.
Depth-First Search (DFS) and Breadth-First Search (BFS) are two fundamental graph traversal algorithms used to explore nodes and edges of a graph. Each algorithm follows a distinct strategy to visit all vertices of a graph.
DFS Strategy: DFS explores a graph by going as deep as possible along each branch before backtracking. This approach uses a stack (LIFO structure) to keep track of the vertices to be explored.
DFS Algorithm: 1. Start at the root (or any arbitrary node). 2. Push the starting node onto the stack. 3. Pop the top node from the stack and mark it as visited. 4. Push all adjacent unvisited nodes of the current node onto the stack. 5. Repeat steps 3 and 4 until the stack is empty.
Example: Consider the following graph:
A - B - D
| |
C EStarting from vertex A, the DFS traversal order would
be: A -> C -> B -> E -> D.
Applications of DFS: - Detecting cycles in a graph - Topological sorting - Solving puzzles like mazes - Pathfinding algorithms
BFS Strategy: BFS explores a graph level by level, visiting all neighbors of a vertex before moving on to the next level. This approach uses a queue (FIFO structure) to keep track of the vertices to be explored.
BFS Algorithm: 1. Start at the root (or any arbitrary node). 2. Enqueue the starting node. 3. Dequeue a node from the front of the queue and mark it as visited. 4. Enqueue all adjacent unvisited nodes of the current node. 5. Repeat steps 3 and 4 until the queue is empty.
Example: Consider the same graph:
A - B - D
| |
C EStarting from vertex A, the BFS traversal order would
be: A -> B -> C -> D -> E.
Applications of BFS: - Finding the shortest path in an unweighted graph - Level order traversal of a tree - Solving problems like the shortest path in a maze - Web crawlers and network broadcasting
DFS goes deep into a graph, while BFS explores each level of a graph.
DFS uses a stack, while BFS uses a queue.
DFS can be implemented using recursion or an explicit stack, while BFS relies on an explicit queue.
BFS is better for finding the shortest path in an unweighted graph, while DFS is useful for tasks like topological sorting and cycle detection.
I hope this gives you a clear understanding of DFS and BFS! Let me know if you need further details or examples.
Let G be a graph and let v be a vertex of G. Say what is meant by the degree of v.
A graph is called k-regular if each of its vertices has degree k. Construct an example of:
a 2-regular graph with 5 vertices,
a 3-regular graph with 6 vertices.
State, without proving, a result connecting the degrees of the vertices of a graph G with the number of its edges.
Use the result in part (c) to find the number of edges of a 3-regular graph with 10 vertices
Explain why it is not possible to construct a 3-regular graph with 9 vertices.
Draw the graph , which has the vertices \(v_1\),\(v_2\),\(v_3\),\(\ldots\),\(v_7\), and the adjacency list:
Given the following definitions for simple, connected graphs:
Draw \(K_4\), \(C_4\), and \(W_4\).