Question

In a simple undirected graph, the sum of the degrees of all vertices is 20. How many edges are present in the graph?

• A. 8
• B. 10
• C. 12
• D. 20

Hint:

For a "simple" graph with $E$ edges, the number of vertices $n$ must satisfy the inequality $E \leq \frac{n(n-1)}{2}$. For 10 edges, the graph must have at least 5 vertices.

Answer 1

The Correct Option is B

Concept: This is a direct application of the Handshaking Lemma, which states that every edge in an undirected graph contributes exactly 2 to the total sum of degrees (once for each of its two endpoints).
• Principle: Total Degree Sum = $2 \times (\text{Total Edges})$

Step 1: Setting up the equation based on given data.
The problem explicitly states that: \[ \sum \text{deg}(v) = 20 \]

Step 2: Using the degree-sum formula.
We use the relationship: \[ 2E = \text{Sum of degrees} \] Substituting the given value: \[ 2E = 20 \]

Step 3: Calculating the final value.
Divide the total sum by 2 to find the number of edges: \[ E = \frac{20}{2} = 10 \] There are 10 edges in the graph.