Let G be a graph with vertex set V = {v1, v2, v3, v4, v5}.
If the degrees of the vertices are 1, 2, 3, 4, 6, respectively, how many edges are in G?
The number of edges
=(sum of degrees of all vertices)/2
MathMate: Is this correct for the following problem:
1 + 2 + 3 + 4 + 6 = 16
2E = 16
E = 8
Does the graph have 8 edges?
To find the number of edges in the graph G, we can use the Handshaking Lemma, also known as the Degree Sum Formula. This lemma states that the sum of the degrees of all vertices in a graph is equal to twice the number of edges.
Mathematically, the Handshaking Lemma can be expressed as:
Σ (degree(v)) = 2 * E,
where Σ (degree(v)) represents the sum of the degrees of all vertices, E represents the number of edges in the graph, and degree(v) denotes the degree of vertex v.
In this case, we are given the degrees of the vertices v1, v2, v3, v4, and v5 as 1, 2, 3, 4, and 6, respectively. Therefore, we can calculate the sum of the degrees as follows:
Σ (degree(v)) = 1 + 2 + 3 + 4 + 6
= 16.
Using the Handshaking Lemma, we have:
Σ (degree(v)) = 2 * E,
16 = 2 * E.
Simplifying the equation, we find:
E = 16 / 2,
E = 8.
Therefore, there are 8 edges in graph G.