True or False: The sum of the entries in the incidence matrix of any graph is equal to twice the number of edges in the graph.

Statement Analysis

The statement asserts that the sum of the entries in the incidence matrix of any graph is equal to twice the number of edges in that graph.

Understanding the Concept

1. Incidence Matrix Definition: The incidence matrix of a graph relates vertices to edges. For a graph with $n$ vertices and $m$ edges, the incidence matrix $A$ is an $n \times m$ matrix where:
   • $A_{ij} = 1$ if vertex $i$ is incident to edge $j$ (i.e., edge $j$ connects vertex $i$), and
   • $A_{ij} = 0$ otherwise.

Analyzing the Sum of Entries

2. Calculating the Sum: In an undirected graph:

   • Each edge connects exactly two vertices.
   • Thus, each edge contributes exactly 2 to the total sum of the entries in the incidence matrix (one for each of the two incident vertices).

3. Total Contribution of Edges: If there are $m$ edges in the graph, the total contribution to the sum of the entries in the incidence matrix is: $\text{Sum of entries} = 2m$ Hence, the sum of all entries in the incidence matrix is indeed twice the number of edges.

Final Answer

Therefore, the statement is True. The sum of the entries in the incidence matrix of any graph is indeed equal to twice the number of edges in the graph.