Free Printable Worksheets for learning Graph theory at the College level

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Graph Theory

Introduction

Graph theory is the mathematical study of networks, including nodes and edges that connect them. It is used in a variety of fields, including computer science, physics, and social sciences. Understanding the basic concepts and properties of graphs is essential to analyzing many real-world problems.

Key Concepts

Graph

A graph is a mathematical structure that consists of a set of vertices (or nodes) and a set of edges that connect them.
A graph is represented using a visual diagram in which vertices are represented by dots or circles and edges are represented by lines that connect them.

Directed and Undirected Graphs

A graph can be directed or undirected.
A directed graph has edges that point in a specific direction, whereas an undirected graph has edges that do not have a direction.

Adjacency Matrix

An adjacency matrix is a way to represent a graph using a matrix where each row and column represents a vertex.
The value in each cell of the matrix represents the presence or absence of an edge between two vertices.

Degrees

The degree of a vertex is the number of edges that connect to it.
In a directed graph, the in-degree of a vertex is the number of edges that point to it, and the out-degree is the number of edges that point away from it.

Path and Cycle

A path is a sequence of vertices in a graph, where each vertex is connected to the next by an edge.
A cycle is a path that starts and ends at the same vertex.

Trees

A tree is a connected graph with no cycles.
It is a special type of graph that is used in computer science and algorithms.

Applications

Graph theory has many applications, including:

Computer networks and the internet
Social networks and communication systems
Transportation networks, such as highways and flight routes
Biological networks, such as protein interactions in the cell

Conclusion

Graph theory is an essential part of many fields and has a wide range of applications. Understanding the key concepts and properties of graphs is essential to solving and analyzing real-world problems.

Here's some sample Graph theory vocabulary lists Sign in to generate your own vocabulary list worksheet.

Word	Definition
Graph	A collection of nodes (also known as vertices) and edges.
Node	A point in a graph.
Edge	A line connecting two nodes in a graph.
Degree	The number of edges connected to a node.
Adjacent	Two nodes that are connected by an edge.
Path	A sequence of connected edges and nodes in a graph.
Cycle	A path that starts and ends at the same node.
Connected	A graph where there is a path from any node to any other node.
Disconnected	A graph where there are two or more disconnected subgraphs.
Weighted Graph	A graph where edges have a numerical weight or cost associated with them.
Directed Graph	A graph where the edges have a direction, indicating a one-way relationship between nodes.
Undirected Graph	A graph where the edges have no direction, indicating a two-way relationship between nodes.
Eulerian Graph	A graph that contains a cycle that passes through every edge exactly once.
Hamiltonian Graph	A graph that contains a path that passes through every node exactly once.
Planar Graph	A graph that can be drawn on a plane without any edges crossing.
Vertex Coloring	A way of coloring the nodes of a graph such that no two adjacent nodes have the same color.
Chromatic Number	The smallest number of colors needed to properly color the nodes of a graph.
Tree	A connected graph with no cycles.
Spanning Tree	A subgraph of a graph that contains all the nodes of the original graph and is also a tree.
Isomorphic	Two graphs are isomorphic if they have the same structure, but the nodes may be labeled differently.

Here's some sample Graph theory study guides Sign in to generate your own study guide worksheet.

Graph Theory Study Guide

Introduction

Graph theory is the mathematical study of objects known as graphs. A graph in graph theory is a collection of vertexes, which are also called points or nodes, and edges, which are lines or arcs that connect any two vertexes. Graph theory has a wide range of real-world applications in areas like computer science, social media analysis, transportation systems, and many others.

Terminology

Vertex/Node/Point: A point in a graph that represents an object or concept.
Edge/Arc/Line: A connection between two vertexes in a graph that represents a relationship between two objects or concepts.
Degree: The number of edges connected to a vertex.
Path: A sequence of edges that connect two vertexes.
Cycle: A path that starts and ends at the same vertex and passes through each vertex exactly once.
Connected Graph: A graph where there is a path between any two vertexes.
Complete Graph: A graph where each vertex is connected to every other vertex.
Subgraph: A subset of a graph's vertices and edges that form a smaller graph themselves.
Directed Graph: A graph where edges have a direction.
Weighted Graph: A graph where edges have weights or values assigned to them.

Common Problems

Shortest Path: Find the shortest path between two vertexes in a graph.
Minimum Spanning Tree: Find the tree that connects all vertexes in a graph, with the least possible total weight.
Maximum Flow: Determine the maximum amount of flow that can go through a graph from a source vertex to a sink vertex.
Planarity Testing: Determine if a graph can be drawn on a plane without any edges crossing.
Vertex Coloring: Assign colors to vertexes in a graph so that no two adjacent vertexes have the same color.
Clique Finding: Find the largest clique, which is a set of vertexes that are all adjacent to each other, in a graph.

Graph Representation

Graphs can be represented using various data structures like adjacency matrix, adjacency list, incidence matrix, and others. Each data structure has its pros and cons and is suitable for different types of graphs and problems.

Conclusion

Graphs are an essential part of modern mathematics with many real-world applications. In graph theory, vertexes represent objects, and edges represent relationships between these objects. You can solve many problems using graph theory, like finding the shortest path or determining the maximum flow. Different data structures can be used to represent graphs, depending on the problem and the graph's characteristics.

Here's some sample Graph theory practice sheets Sign in to generate your own practice sheet worksheet.

Graph Theory Practice Sheet

Problem 1

Given a simple graph G = (V, E) with n vertices and m edges, prove that m <= n(n-1)/2.

Problem 2

For any tree T with at least two vertices, prove that there exist at least two vertices in T that have the same degree.

Problem 3

If a graph G is connected and has n vertices, prove that the size of the smallest edge cut in G is at least n-1.

Problem 4

Let G be a graph with n vertices and k components. Prove that G has at least n-k edges.

Problem 5

Consider a complete graph Kn on n vertices. What is the chromatic number of Kn?

Problem 6

Let G = (V, E) be a graph. If every vertex in G has degree at least k, prove that G contains a path of length at least k.

Problem 7

Let G be a simple undirected graph with n vertices and m edges. Prove that if m > (n-1)(n-2)/2, then G contains a triangle.

Problem 8

A bipartite graph G = (U, V, E) with |U| = |V| = n is called a König graph if it has a perfect matching. Prove that a König graph has a vertex cover of size n.

Problem 9

Let G be a graph with n vertices and k connected components, such that each component has an odd number of vertices. Prove that the number of odd-degree vertices in G is even.

Problem 10

Let G be a graph with n vertices and m edges. Prove that the complement of G has at least n(n-1)/2 - m edges.

Good Luck!

Practice Sheet for Graph Theory

What is a graph?
What is an undirected graph?
What is a directed graph?
What is the difference between an undirected graph and a directed graph?
What is a path in a graph?
What is the degree of a vertex in a graph?
What is a cycle in a graph?
What is a connected graph?
What is a disconnected graph?
What is a bipartite graph?
What is a complete graph?
What is a tree?
What is an Eulerian path?
What is a Hamiltonian path?
What is an adjacency matrix?
What is an adjacency list?
What is the shortest path problem?
What is the maximum flow problem?
What is the minimum spanning tree problem?
What is the traveling salesman problem?

Graph Theory Practice Sheet

Task 1

Create a graph with the following properties: - 5 vertices - 4 edges

Task 2

Given the following graph, determine the degree of vertex A:

A -- B -- C \ / \ / \ / \ / D

Task 3

Given the following graph, determine if it is a complete graph:

A -- B -- C \ / \ / \ / \ / D

Task 4

Given the following graph, determine if it is a tree:

A -- B -- C \ / \ / \ / \ / D

Task 5

Given the following graph, determine the number of edges:

A -- B -- C \ / \ / \ / \ / D

Task 6

Given the following graph, determine the number of connected components:

A -- B -- C \ / \ / \ / \ / D

Task 7

Given the following graph, determine if it is a bipartite graph:

A -- B -- C \ / \ / \ / \ / D

Here's some sample Graph theory quizzes Sign in to generate your own quiz worksheet.

Problem	Answer
Define a simple graph	A graph that has no self-loops or multiple edges
What is a degree in a graph?	The degree of a vertex is the number of edges that are incident on it
Can a graph with an odd number of vertices have all vertices of even degree?	No
What is an Eulerian graph?	A graph that contains an Eulerian cycle
What is a Hamiltonian graph?	A graph that contains a Hamiltonian cycle
What is a bipartite graph?	A graph in which the vertices can be partitioned into two sets such that no two vertices within the same set share an edge
What is a complete graph?	A graph in which each vertex is adjacent to every other vertex
Define a planar graph	A graph that can be drawn in the plane without any edges crossing
According to the Four Color Theorem, what is the minimum number of colors that can be used to color any planar map?	Four
What is a spanning tree?	A subgraph that is a tree and includes every vertex of the original graph

Question	Answer
What is a graph?	A graph is a mathematical structure used to represent relationships between objects. It is composed of vertices (or nodes) which are connected by edges.
What are the two types of graphs?	The two types of graphs are directed and undirected graphs.
What is an example of a directed graph?	A directed graph is a graph in which the edges have a direction associated with them. An example of a directed graph is a map of a city, where the edges represent roads and the direction of the edge indicates the direction of travel.
What is an example of an undirected graph?	An undirected graph is a graph in which the edges have no direction associated with them. An example of an undirected graph is a social network, where the edges represent connections between people.
What is a path in a graph?	A path in a graph is a sequence of edges that connects two vertices.
What is a cycle in a graph?	A cycle in a graph is a path that starts and ends at the same vertex.
What is the degree of a vertex?	The degree of a vertex is the number of edges incident to that vertex.
What is an adjacency matrix?	An adjacency matrix is a matrix representation of a graph, where the rows and columns represent the vertices and the entries in the matrix indicate whether or not there is an edge between two vertices.
What is an adjacency list?	An adjacency list is a list representation of a graph, where each vertex is associated with a list of its adjacent vertices.
What is the degree sequence of a graph?	The degree sequence of a graph is the sequence of degrees of the vertices in the graph.

Graph Theory Quiz

Question	Answer
What is a graph?	A graph is a collection of points, called vertices, and lines connecting them, called edges.
What are the two types of graphs?	The two types of graphs are directed graphs and undirected graphs.
What is a directed graph?	A directed graph is a graph in which edges have a direction associated with them, indicating a one-way relationship between the two vertices.
What is an undirected graph?	An undirected graph is a graph in which edges do not have a direction associated with them, indicating a two-way relationship between the two vertices.
What is an adjacency matrix?	An adjacency matrix is a matrix that represents the connections between vertices in a graph. Each row and column of the matrix represents a vertex, and the value at each intersection indicates whether there is an edge between the two vertices.
What is an adjacency list?	An adjacency list is a list of all the vertices adjacent to a given vertex. Each element in the list contains the other vertex that is adjacent to the given vertex and the edge connecting them.
What is a weighted graph?	A weighted graph is a graph in which each edge has an associated weight or cost. This weight is used to represent the cost of traversing the edge.
What is the shortest path problem?	The shortest path problem is the problem of finding the shortest path between two vertices in a graph.
What is the minimum spanning tree problem?	The minimum spanning tree problem is the problem of finding the cheapest way to connect all the vertices in a graph.
What is the maximum flow problem?	The maximum flow problem is the problem of finding the maximum amount of flow that can be sent through a network of edges in a graph.