The column sum in an incidence matrix for a directed graph having no self loop is __________?Group of answer choices012equal to the number of edges

To determine the column sum in an incidence matrix for a directed graph with no self-loop, let's analyze the concepts involved:

Understanding the Incidence Matrix

1. Definition: An incidence matrix for a directed graph is a matrix that represents the relationship between vertices and edges. In this matrix, rows correspond to vertices, and columns correspond to directed edges.

2. Column Entries: Each entry in the matrix is either:

   • 1 if the vertex is the starting (tail) vertex of the directed edge,
   • -1 if the vertex is the ending (head) vertex of the directed edge,
   • 0 if the vertex is not incident to that edge.

Analysis of Column Sums

1. Column Sum Interpretation: For any directed edge, the tail vertex contributes 1 to the column sum, while the head vertex contributes -1. Thus, for each edge, the contributions will sum up depending on how many heads and tails the edge connects.

2. Directed Graph with No Self-Loops: In this case, since there are no edges that start and end at the same vertex (i.e., no self-loops), each edge can be represented by precisely one 1 and one -1 in the incidence matrix. The contribution of each edge cancels out, resulting in a column sum of 0.

Conclusion

Therefore, for a directed graph with no self-loops, the column sum in the incidence matrix will always be zero.

Final Answer

The column sum in an incidence matrix for a directed graph having no self-loop is 0.