Is it possible for the degrees of the vertices to be 3, 6, 2, 1, 5, respectively? Why or why not?

Question

Let G be a graph with vertex set V = {v1, v2, v3, v4, v5}.

Answer 1

The sum of the degrees is 17 which is an odd number. The sum of the degrees is twice the number of edges, so it is not possible, even with loops, to have an odd sum. The reason for this is that an edge has two endpoints with a vertex at each end. Also a loop increases the degree of a vertex by 2, an even number.