The handshaking lemma says the degrees of all vertices sum to twice the number of edges. Why must the total be even, and what does that immediately rule out?

D. Because each edge adds 2 to the sum, so the degree sum is always even.

Each edge has two endpoints and adds 1 to each, contributing exactly 2, so the sum is always even whatever the graph looks like. Odd numbers sum to an even total only in even quantity, so a graph with exactly three odd-degree vertices does not exist. It underlies Euler's theorem: a closed circuit needs every degree even, an open trail exactly two odd, never one.