Question

If \(G\) is a graph without loops, what can you say about the sum of the entries in (i) any row or column of the adjacency matrix of \(G\) ? (ii) any row of the incidence matrix of \(G\) ? (iii) any column of the incidence matrix of \(G\) ?

Short answer

The sum of the entries in (i) any row or column of the adjacency matrix of \(G\) gives the degree of the corresponding vertex, (ii) any row of the incidence matrix of \(G\) shows the degree of the corresponding vertex, and (iii) in any column of the incidence matrix of \(G\), the sum will always be 2 as it counts the vertices connected by the linked edge.

Step-by-step solution

Analyze Entries of the Adjacency Matrix

Any row or column of the adjacency matrix of \(G\) corresponds to a vertex in the graph. The entries in this row or column denote the number of edges it shares with every other vertex. Therefore, the sum of those entries would be the degree of the vertex represented by that row or column. This is because each entry is an edge to another vertex, and the sum of all entries, thus, gives all the edges of that vertex.

Analyze Row Entries of the Incidence Matrix

In the incidence matrix, each row corresponds to a vertex, and the entry is 1 if the vertex is an end vertex of the corresponding edge, 0 otherwise. Hence, summing the row entries gives the degree of the vertex as it is counting the number of edges the vertex is connected to.

Analyze Column Entries of the Incidence Matrix

Each column in the incidence matrix represents an edge of the graph. The entries in a column will be 1 for the two vertices which the edge connects. Therefore, the sum of entries in any column of the incidence matrix will always be 2, as an edge is connected to exactly two vertices.

Key concepts

Adjacency Matrix

An adjacency matrix is a powerful way to represent a graph using a square matrix. In this matrix:

• Each row and column correspond to a vertex of the graph.
• The entry at row i and column j is 1 if there is an edge between vertex i and vertex j, and 0 otherwise.

Since graph \(G\) has no loops, each diagonal entry of the adjacency matrix is 0, meaning a vertex does not connect to itself. The sum of the entries in any row or column of the adjacency matrix gives the number of edges that are incident to the vertex represented by that row or column. Therefore, this sum is the degree of that vertex.

Incidence Matrix

The incidence matrix is another way to represent a graph, focusing on the relationship between vertices and edges. Here's how it works:

• Each row represents a vertex.
• Each column represents an edge.
• The entry is 1 if the vertex is an endpoint of the edge, and 0 otherwise.

For graphs without loops, the arrangement of 1s in the matrix shows which vertices are endpoints for the edges. This structure easily helps to see the connections in the graph. The incidence matrix provides a direct way to calculate the degree of each vertex and easily identify the endpoints for each edge.

Vertex Degree

The vertex degree is an essential concept in understanding the structure of a graph. It is defined as:

• The number of edges incident to a vertex.
• In undirected graphs without loops, each edge contributes 1 to the degree count of each connected vertex.

In the context of an adjacency matrix, the degree of a vertex is the sum of its row or column entries. In terms of the incidence matrix, it is the sum of the entries in the corresponding row. The vertex degree helps in analyzing the connectivity and reachability within the graph.

Edges

Edges are the connections between vertices in a graph. They can be:

• Represented in a graph with a pair of vertices \((u, v)\).
• Visualized in an adjacency matrix by the non-zero off-diagonal entries.
• Identified in an incidence matrix by columns that contain 1s at positions corresponding to the vertices they connect.

In a simple graph like \(G\), which has no loops, each edge connects exactly two different vertices. Understanding how edges are represented in different matrix forms of a graph is crucial for graph analysis.

Graphs without Loops

Graphs without loops are ones where no vertex connects to itself. This characteristic simplifies the adjacency and incidence matrices:

• The diagonal entries of the adjacency matrix are all 0 because there are no self-connections.
• Each column in the incidence matrix sums to 2 because every edge connects exactly two different vertices.

This simplicity ensures that calculations involving sums of matrix entries (like row or column totals) reflect actual inter-vertex connections, not self-loop artifacts. While analyzing such graphs, we ensure that all vertices interact only through distinct edges connecting them to other vertices.