Final answer:
The sum of the entries in a column of the adjacency matrix for an undirected graph represents the degree of each vertex in the graph.
Step-by-step explanation:
The sum of the entries in a column of the adjacency matrix for an undirected graph represents the degree of each vertex in the graph. The degree of a vertex is the number of edges incident to it. To find the sum, we add up all the entries in the column, including the diagonal element if there is a self-loop.
For example, consider a simple undirected graph with four vertices:
A | B | C | D
A | 0 | 1 | 0 | 1
B | 1 | 0 | 1 | 0
C | 0 | 1 | 0 | 1
D | 1 | 0 | 1 | 0
The sum of the entries in the first column is 0+1+0+1 = 2. This means vertex A has a degree of 2.
In general, the sum of the entries in any column or row of the adjacency matrix is equal to twice the number of edges in the graph. This is because each edge is represented by two entries in the matrix, one for each endpoint vertex.