Sum of Degrees of Vertices Theorem

The Sum of Degrees of Vertices Theorem (also known as the Handshaking Lemma) is a fundamental result in graph theory that relates the sum of all vertex degrees to the number of edges in a graph.

The Theorem

If G = (V, E) is a non-directed graph with vertices V = {V₁, V₂, …, V_n}, then −

∑_i=1ⁿ deg(V_i) = 2|E|

This is because each edge contributes exactly 2 to the total degree sum − one for each of its endpoints.

Example

Corollary 1: Directed Graphs

If G = (V, E) is a directed graph with vertices V = {V₁, V₂, …, V_n}, then −

∑_i=1ⁿ deg⁺(V_i) = |E| = ∑_i=1ⁿ deg⁻(V_i)

In a directed graph, each edge contributes 1 to the outdegree of its source and 1 to the indegree of its destination. So the total indegree equals the total outdegree, and both equal |E|.

Corollary 2: Odd Degree Vertices

In any non-directed graph, the number of vertices with odd degree is always even.

This follows because the total degree sum must be even (2|E|), so the odd-degree contributions must cancel out, requiring an even count of such vertices.

Corollary 3: Regular Graphs

In a non-directed graph where the degree of each vertex is exactly k (a k-regular graph) −

k × |V| = 2|E|

Corollary 4: Minimum Degree Bound

In a non-directed graph where the degree of each vertex is at least k −

k × |V| ≤ 2|E|

Corollary 5: Maximum Degree Bound

In a non-directed graph where the degree of each vertex is at most k −

k × |V| ≥ 2|E|

Conclusion

The Sum of Degrees theorem states that the total degree in any undirected graph is exactly twice the number of edges. This fundamental identity leads to important corollaries about odd-degree vertices, regular graphs, and degree bounds in both directed and undirected graphs.