Question

Draw the graphs whose incidence matrices are given below: \(\begin{array}{llllllllllll}\text { (1) } & \mathrm{e}_{1} & \mathrm{e}_{2} & \mathrm{e}_{3} & \mathrm{e}_{4} & \mathrm{e}_{5} & \mathrm{e}_{6} & \mathrm{e}_{7} & \mathrm{e}_{8} & \mathrm{e}_{9} & \mathrm{e}_{10} & \mathrm{e}_{11} & \mathrm{e}_{12}\end{array}\) \(\begin{array}{lllllllllllll}\mathrm{v}_{1} & \mid 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ \mathrm{v}_{2} & 10 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ \mathrm{v}_{3} & 10 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \mid\end{array}\) \(\mathrm{A}=\mathrm{v}_{4} \mid \begin{array}{llllllllllll}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\) \(\begin{array}{lllllllllllll}\mathrm{v}_{5} & \mid 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \mid \\ \mathrm{v}_{6} & \mid 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \mid \\ \mathrm{v}_{7} & \mid 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \mid\end{array}\) \(\begin{array}{lllllllllll}\text { (2) } & \mid 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ & \mid 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\ \mathrm{~B}= & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ & \mid 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ & \mid 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0\end{array}\)

Short answer

The graphs for the given incidence matrices can be drawn as follows: Graph 1: ``` ⌂---ₑ₄--►◯---ₑ₅--►◉---ₑ₆--►◯ │ ⌆ │ ⌆ │ ₑ₁ ₑ₈ ₑ₂ ₑ₉ ₑ₃ ⌦ │ ⌦ │ ⌦ ↓ ⌆ ↓ ⌆ ↓ ○---ₑ₁₂--►◎---ₑ₁₀--►◒---ₑ₁₁--►∅ ``` Graph 2: ``` ◉---ₑ₂--►◊---ₑ₇--►⌇ │ ⌆ │ ₑ₁ ₑ₈ ₑ₁₀ │ ⌆ │ ◕---ₑ₄--►₍₎---ₑ₉--►◓ ⌆ ₑ₅ ⌆ ┘ ```

Step-by-step solution

Graph 1: Analyzing Incidence Matrix A

: First, we will analyze the incidence matrix A: $\begin{array}{lllllllllllll}\mathrm{v}_{1} & \mid 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ \mathrm{v}_{2} & 10 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ \mathrm{v}_{3} & 10 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \mid\end{array}$ $\mathrm{A}=\mathrm{v}_{4} \mid \begin{array}{llllllllllll}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}$ $\begin{array}{lllllllllllll}\mathrm{v}_{5} & \mid 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \mid \\ \mathrm{v}_{6} & \mid 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \mid \\ \mathrm{v}_{7} & \mid 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \mid\end{array}$ This matrix has 7 vertices and 12 edges. We can now draw the graph representing these connections.

Graph 1: Drawing the Graph

: To draw the graph, we follow the connections present in the incidence matrix A. The following graph represents the incidence matrix A: ``` ⌂---ₑ₄--►◯---ₑ₅--►◉---ₑ₆--►◯ │ ⌆ │ ⌆ │ ₑ₁ ₑ₈ ₑ₂ ₑ₉ ₑ₃ ⌦ │ ⌦ │ ⌦ ↓ ⌆ ↓ ⌆ ↓ ○---ₑ₁₂--►◎---ₑ₁₀--►◒---ₑ₁₁--►∅ ``` Vertices: v1 (⌂), v2 (◯), v3 (◉), v4 (○), v5 (◎), v6 (◒), v7 (∅) Edges: e1 to e12

Graph 2: Analyzing Incidence Matrix B

: Next, we will analyze the incidence matrix B: $\begin{array}{lllllllllll}\text { (2) } & \mid 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ & \mid 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\ \mathrm{~B}= & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ & \mid 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ & \mid 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0\end{array}$ This matrix has 5 vertices and 10 edges. We can now draw the graph representing these connections.

Graph 2 : Drawing the Graph

: To draw the graph, we follow the connections present in the incidence matrix B. The following graph represents the incidence matrix B: ``` ◉---ₑ₂--►◊---ₑ₇--►⌇ │ ⌆ │ ₑ₁ ₑ₈ ₑ₁₀ │ ⌆ │ ◕---ₑ₄--►₍₎---ₑ₉--►◓ ⌆ ₑ₅ ⌆ ┘ ``` Vertices: v1 (◉), v2 (◊), v3 (⌇), v4 (₍₎), v5 (◕) Edges: e1 to e10 That's it. We have successfully drawn the graphs as per their incidence matrices.

Key concepts

Graph Theory

Graph theory is a fundamental field within Discrete Mathematics, focusing on the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph is made up of vertices (also called nodes or points) connected by edges (also called links or lines). Applications of graph theory are pervasive and include areas such as computer science, biology, sociology, and urban planning.

Graphs come in different types, such as simple graphs, directed graphs, weighted graphs, and so on. Simple graphs consist of vertices connected by non-directional edges where no loops or multiple edges between the same pair of vertices are allowed. Directed graphs, or digraphs, have edges with a direction, indicating a one-way relationship between two vertices. Understanding the properties of these graphs is crucial for creating algorithms that can traverse, search, or optimize paths within different networks.

In the context of the provided exercise, the incidence matrices represent a specific type of graph. Incidence matrices are binary matrices that show the relationship between vertices and edges in a graph. They are particularly useful for computational and theoretical analysis of graph structures. By interpreting these matrices, it's possible to visualize the architecture of the graph they represent.

Drawing Graphs from Matrices

Drawing graphs from matrices, specifically incidence matrices, is a skill often utilized in graph theory. An incidence matrix is an 'n x m' matrix where 'n' represents the number of vertices and 'm' denotes the number of edges. In an incidence matrix, each row corresponds to a vertex and each column to an edge. The entries in the matrix are either 0 or 1. A '1' indicates that the vertex corresponding to the row is incident to the edge corresponding to the column, meaning the vertex is one of the ends of that edge.

To draw a graph from such a matrix, one should follow these steps: Identify the vertices and label them, typically as v1, v2, v3, and so on. Then, analyze each column to determine which vertices are connected by the respective edge. Each '1' in the column signifies that the edge connects two vertices. Next, draw the vertices as points in a two-dimensional space and connect them with lines representing the edges, based on the connections outlined in the incidence matrix.

Exercise Improvement Advice

When drawing graphs from incidence matrices, it's easy to get overwhelmed if the matrix is large. To avoid confusion, one can use different colors or labels to distinguish the edges while drawing. Additionally, starting with the vertex that has the least or the most connections might help to gradually develop the overall structure without missing any edges. Also, ensuring that no two edges cross each other can aid in achieving a clear representation.

Discrete Mathematics

Discrete Mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. This means that it deals with countable, distinct objects. It encompasses an array of topics such as logic, set theory, combinatorics, graph theory, and algorithms. Within discrete mathematics, graph theory stands out as a significant area of interest that enables us to model and analyze relationships in a variety of systems.

Discrete math forms the backbone of computer science and information theory. It provides the tools for designing data structures, optimizing algorithms, and formulating cryptographic systems. Given its discrete nature, it is particularly important when dealing with digital data. Various algorithms used for searching, sorting, and network flow optimization are conceptualized using principles from discrete mathematics. This field is fundamental for students aiming to delve into computer science, electrical engineering, or further studies in applied mathematics.

In this context, understanding discrete mathematics is essential for interpreting incidence matrices and drawing accurate graphs from them. The ability to transform a matrix into a graph requires a foundational grasp of discrete structures and how they relate to visual and analytical representations in graph theory.