If G is a graph with e edges and n vertices, the sum of the degrees of all…

To solve this problem, we apply the Handshaking Lemma from graph theory. This fundamental theorem states that in any undirected graph, the sum of the degrees of all vertices is exactly twice the number of edges. Each edge connects two vertices, contributing 1 to the degree count of each endpoint. Therefore, every single edge adds exactly 2 to the total sum of degrees across the entire graph.\nIf a graph G has 'e' edges, we multiply this count by 2 to find the total degree sum. Mathematically, Sum of Degrees = 2 * e.\nOption A (e) is incorrect because it ignores that each edge contributes to two vertices, not one. Option B (e/2) is incorrect as it suggests the sum is less than the number of edges, which contradicts the definition of degree. Option C (e 2) is mathematically invalid notation in this context. Thus, the correct answer is Option D (2e).