

# Representation of graphs: adjacency matrix and adjacency list



An adjacency matrix is a square matrix where the element in row $i$ and column $j$ is $1$ if vertices $i$ and $j$ are connected by an edge, and $0$ otherwise. This representation is efficient for dense graphs, where most vertices have many edges.

Consider the following graph with vertices $A, B, C, D, E$ and edges $AB, AC, AD, AE, BC, BD, BE, CD, CE, DE$.

The adjacency matrix representation of this graph is:

$$
\begin{bmatrix}
0 & 1 & 1 & 1 & 1 \\
1 & 0 & 1 & 0 & 1 \\
1 & 1 & 0 & 1 & 1 \\
1 & 0 & 1 & 0 & 1 \\
1 & 1 & 1 & 1 & 0
\end{bmatrix}
$$

An adjacency list is a collection of lists, where each list represents a vertex and contains the vertices it is connected to by an edge. This representation is efficient for sparse graphs, where most vertices have few edges.

Using the same graph as before, the adjacency list representation is:

$$
\begin{aligned}
A &= \{B, C, D, E\} \\
B &= \{A, C, D, E\} \\
C &= \{A, B, D, E\} \\
D &= \{A, B, C, E\} \\
E &= \{A, B, C, D\}
\end{aligned}
$$

## Exercise
Consider the following graph with vertices $A, B, C, D, E$ and edges $AB, AC, AD, AE, BC, BD, BE, CD, CE, DE$.

1. Convert this graph to an adjacency matrix representation.
2. Convert this graph to an adjacency list representation.

# Graph coloring and its applications

Graph coloring is the process of assigning colors to the vertices of a graph in such a way that no two adjacent vertices share the same color. This is an important concept in graph theory, as it has applications in computer science, combinatorics, and more.

A graph is said to be $k$-colorable if it can be colored using at most $k$ colors. The chromatic number of a graph is the smallest $k$ for which the graph is $k$-colorable.

Graph coloring has many applications in computer science, such as register allocation in compilers and scheduling in operating systems.

## Exercise
Consider the following graph with vertices $A, B, C, D, E$ and edges $AB, AC, AD, AE, BC, BD, BE, CD, CE, DE$.

1. Determine the chromatic number of this graph.
2. Find a proper coloring of the graph with the smallest number of colors.

# Euler's formula and its connection to graph coloring

Euler's formula is a fundamental result in graph theory that relates the number of vertices, edges, and faces in a planar graph. It states that for any planar graph $G$ with $V$ vertices, $E$ edges, and $F$ faces, the following equation holds:

$$
V - E + F = 2
$$

Euler's formula can be used to prove that a graph is $4$-colorable if and only if it satisfies the conditions of Euler's formula.

## Exercise
Consider the following graph with vertices $A, B, C, D, E$ and edges $AB, AC, AD, AE, BC, BD, BE, CD, CE, DE$.

1. Determine the number of vertices, edges, and faces in this graph.
2. Use Euler's formula to determine if this graph is $4$-colorable.

# Hamiltonian paths and cycles in graphs

A Hamiltonian path in a graph is a path that visits each vertex exactly once and returns to the starting vertex. A Hamiltonian cycle is a Hamiltonian path that also includes the starting vertex.

Determining the existence of Hamiltonian paths and cycles in a graph is a well-known NP-complete problem, meaning that finding an efficient algorithm for solving it is an open question in computer science.

## Exercise
Consider the following graph with vertices $A, B, C, D, E$ and edges $AB, AC, AD, AE, BC, BD, BE, CD, CE, DE$.

1. Determine if this graph contains a Hamiltonian path.
2. Determine if this graph contains a Hamiltonian cycle.

# Planar graphs and their properties

A planar graph is a graph that can be embedded in the plane such that no two edges intersect, except at their common vertices. Planar graphs have many interesting properties, such as being $4$-colorable and having a unique embedding up to isomorphism.

Determining if a graph is planar is a well-known NP-complete problem, meaning that finding an efficient algorithm for solving it is an open question in computer science.

## Exercise
Consider the following graph with vertices $A, B, C, D, E$ and edges $AB, AC, AD, AE, BC, BD, BE, CD, CE, DE$.

1. Determine if this graph is planar.
2. If it is planar, draw a planar embedding of the graph.

# Drawing planar graphs using Euler's formula

Euler's formula can be used to draw a planar embedding of a graph. The process involves repeatedly selecting a face, subdividing it into smaller faces, and connecting the vertices of the original face in a clockwise or counterclockwise direction.

This process ensures that no two edges intersect, except at their common vertices.

## Exercise
Consider the following graph with vertices $A, B, C, D, E$ and edges $AB, AC, AD, AE, BC, BD, BE, CD, CE, DE$.

1. Determine the number of vertices, edges, and faces in this graph.
2. Use Euler's formula to draw a planar embedding of the graph.

# Vertex and edge connectivity in graphs

Vertex connectivity and edge connectivity are two important concepts in graph theory that describe the minimum number of vertices or edges that must be removed to disconnect a graph.

A graph is said to be $k$-vertex-connected if removing any $k-1$ vertices does not disconnect the graph. Similarly, a graph is said to be $k$-edge-connected if removing any $k-1$ edges does not disconnect the graph.

## Exercise
Consider the following graph with vertices $A, B, C, D, E$ and edges $AB, AC, AD, AE, BC, BD, BE, CD, CE, DE$.

1. Determine the vertex connectivity and edge connectivity of this graph.
2. Find a vertex or edge cut of size $k$ for each $k \le 5$.

# Graph drawing algorithms and their complexity

There are many algorithms for drawing graphs, each with its own complexity. Some of the most well-known algorithms include the straight-line drawing algorithm, the layered drawing algorithm, and the circular drawing algorithm.

These algorithms have different time and space complexities, depending on the input graph and the desired output.

## Exercise
Consider the following graph with vertices $A, B, C, D, E$ and edges $AB, AC, AD, AE, BC, BD, BE, CD, CE, DE$.

1. Determine the straight-line drawing of this graph.
2. Determine the layered drawing of this graph.
3. Determine the circular drawing of this graph.

# Applications of graph drawing in computer science and engineering

Graph drawing has many applications in computer science and engineering, such as visualizing data structures, optimizing algorithms, and designing computer networks.

In computer science, graph drawing algorithms are used to visualize graph data structures, such as those used in social networks, web pages, and computer programs.

In engineering, graph drawing algorithms are used to optimize algorithms, such as those used in scheduling tasks or assigning resources in a computer network.

## Exercise
Consider the following graph with vertices $A, B, C, D, E$ and edges $AB, AC, AD, AE, BC, BD, BE, CD, CE, DE$.

1. Describe how this graph can be used to visualize a social network.
2. Describe how this graph can be used to optimize a scheduling algorithm.

# Graph drawing and its connection to other areas of mathematics

Graph drawing has connections to many other areas of mathematics, such as graph theory, combinatorics, and topology.

For example, graph coloring is related to the chromatic number of a graph, which is a topic in graph theory. Additionally, Euler's formula is a fundamental result in topology, as it relates to the number of vertices, edges, and faces in a planar graph.

## Exercise
Consider the following graph with vertices $A, B, C, D, E$ and edges $AB, AC, AD, AE, BC, BD, BE, CD, CE, DE$.

1. Determine the chromatic number of this graph using Euler's formula.
2. Determine the number of vertices, edges, and faces in this graph using Euler's formula.

# Challenges and future directions in graph drawing research

Graph drawing is a complex and challenging field of research, with many open questions and unsolved problems. Some of the current challenges and future directions in graph drawing research include:

- Developing efficient algorithms for drawing large graphs.
- Designing graph drawing algorithms that are robust to errors and noise in the input graph.
- Investigating the relationship between graph drawing and other areas of mathematics, such as geometry and topology.
- Applying graph drawing techniques to real-world problems, such as network design, data visualization, and algorithm optimization.

## Exercise
Consider the following graph with vertices $A, B, C, D, E$ and edges $AB, AC, AD, AE, BC, BD, BE, CD, CE, DE$.

1. Describe a potential application of this graph in a real-world problem.
2. Discuss a potential future direction in graph drawing research.