Question

Draw the graph whose adjacency matrix is given in Fig. \(1.29\). $$ \left(\begin{array}{lllll} 0 & 1 & 1 & 2 & 0 \\ 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 1 \\ 2 & 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \end{array}\right) $$

Short answer

The graph would have 5 vertices and 10 edges. Specifically, the edges would be as follows: two edges between vertices 1 and 4, one edge between vertices 1 and 2, one edge between vertices 1 and 3, one edge between vertices 2 and 5, one edge between vertices 3 and 4, one edge between vertices 3 and 5, and one edge between vertices 4 and 5.

Step-by-step solution

Understanding the matrix

Inspect the given matrix. The value at the \(i^{th}\) row and \(j^{th}\) column in the matrix represent the number of edges between vertices \(i\) and \(j\) in the graph. So, for example, the value of 2 in the first row and fourth column of our matrix represents two edges between vertex 1 and vertex 4.

Plot the edges

Start plotting the graph by connecting the vertices according to the information from the matrix. According to our matrix, we will have an edge between vertices 1 and 2, 1 and 3, and two edges between vertices 1 and 4. Do this for each row in the matrix.

Verify the graph

After finishing plotting all the edges, check if the graph corresponds to the given matrix. This can be done by checking the degree of each vertex in the graph and cross-checking that against the matrix. The degree of a vertex is the total number edges connected to it. For instance, vertex 1 has a total of 4 edges connected to it (1 edge to vertex 2, 1 edge to vertex 3, 2 edges to vertex 4), and this should equal the total of the entries in row 1 of the matrix.

Key concepts

Graph Theory Essentials

Graph theory is a significant area of mathematics and computer science that involves the study of graphs, which models relationships between objects. A graph consists of points called vertices (or nodes) and lines called edges that connect these vertices. The basic premise is simple yet powerful: vertices represent items or positions, and edges represent the connections or paths between them.

The adjacency matrix, like the one given in the exercise, is a core concept in graph theory and provides a way to represent which vertices in a graph are adjacent to each other—that is, directly connected by an edge. The matrix is square, with rows and columns corresponding to the vertices, and the elements of the matrix indicate the number of edges connecting the vertices.

Plotting Graphs with Adjacency Matrices

Plotting a graph from an adjacency matrix requires converting this numerical representation into a visual one. It's like translating a code into a picture. Each row or column in the matrix corresponds to a vertex in your graph. If the matrix entry \( a_{ij} \) is nonzero, that means you have one or more edges between vertices \( i \) and \( j \). The actual number \( a_{ij} \) indicates how many edges you should draw connecting these two vertices.

When plotting, you'll want to space your vertices out evenly and draw smooth, clear lines for edges. You can represent vertices with dots or circles and label them to match their corresponding rows and columns in the matrix. Remember that the position of vertices on the page isn’t fixed; you can arrange them in a way that makes the graph easy to understand and follows the connections indicated by the matrix.

Understanding Graph Vertices and Edges

In any graph, the vertices are the fundamental units, and edges are what create the network between them. They form the backbone of the graph's structure. Vertices can represent anything from cities in a network of roads (edges) to individuals in social networks (where edges signify relationships).

An edge can be directed—showing the direction of a relationship, like an arrow pointing from a follower to a followed person on Instagram—or undirected, where the relationship goes both ways, like a two-way street. In the case of our exercise, the adjacency matrix represents an undirected graph, as seen by the symmetry of the matrix. The fact that there's a '2' in the matrix implies the presence of multiple (specifically, two) edges between a pair of vertices, indicating perhaps a stronger or multi-faceted relationship.

Determining Vertex Degree

The degree of a vertex in a graph is the number of edges that are incident to it. For undirected graphs, counting the degree is straightforward: just total the number of lines touching the vertex. In a directed graph, we might distinguish between the 'in-degree' (number of incoming edges) and 'out-degree' (number of outgoing edges).

In the context of our adjacency matrix, you can find the degree of vertex \( i \) by summing up all the entries in the \( i^{th} \) row or column (since the graph is undirected, both should give the same result). It's a simple yet elegant way to ensure that the graph you've plotted correctly matches the data in the matrix. Vertex degrees are a fundamental measure in graph theory, revealing vital information about the connectivity and centrality of a vertex within the graph.