The sum of degrees of all vertices on a graph might be equal to 5.

Question

The sum of degrees of all vertices on a graph might be equal to 5. i dont know

Answer 1

It can't be 5.

 

 

The sum of the degrees of all vertices in any graph is 2E. Here, E is the number of edges in the graph.

 

To prove this use induction:

 

Base case:

Empty graphs have no edges so the degree sum is 0.

 

Induction:

Assume for some E ≥ 0 that all graphs having E edges have degree sum of 2E. Any graph with E+1 edges is formed by adding a new edge to one with E edges. But that edge connects two vertices so those two vertices have their degrees increase by 1 each while the degrees of all other vertices are unchanged. This adds 2 to the degree sum making it be 2E+2=2(E+1) as desired.

 

Q.E.D.