What is the largest number of vertices in a graph with 35 edges is all vertices are of degree at least 3?

In a graph, the sum of the degrees of all vertices is equal to twice the number of edges. This is known as the Handshaking Lemma. Therefore, if all vertices in a graph with 35 edges have a degree of at least 3, the sum of the degrees of all vertices must be at least 3 times the number of vertices. Since each edge contributes 2 to the sum of degrees, we have 2 * 35 = 3 * V, where V is the number of vertices. Solving for V, we get V = 70/3 = 23.33. Since the number of vertices must be a whole number, the largest possible number of vertices in this graph is 23.