What is The First Theorem of Graph Theory? This theorem gets its name from the fact that it is often the first theorem encountered when one is learning graph theory. The first theorem tells us that if a graph G has size m, then the sum of the degrees of all vertices of G is equal to 2m.

Recall that the size of a graph is the number of edges it has. The degree of a vertex is the number of edges incident to the vertex. So the theorem tells us that if a graph has m edges, then the sum of the degrees of all vertices in the graph is twice m, or 2m. If you think about this for a minute, it should seem pretty intuitive.

Each edge in a graph contributes 1 to the degree count of two different vertices, because each edge is incident to two vertices. So, each edge contributes 2 to the total degree count of all vertices in the graph, so the total degree count is two times the number of edges, or 2m, where m is the size of the graph.

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