Number of Edges in a Regular Graph Degree 46, 8 Vertices | AI Research

Number of Edges in a Regular Graph (Degree 46, 8 Vertices)

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Sep 12, 2026

Regular Graphs & Handshaking Lemma (Edges from Degrees)

In an undirected simple graph, every edge contributes $1$ to the degree of each of its two endpoints. This yields the Handshaking Lemma: $\sum_{v\in V}\deg(v)=2|E|$ .

For a regular graph of degree $k$ on $n$ vertices (a [ $k$ -regular graph]{def="A regular graph where each vertex has degree k"}), each of the $n$ vertices has degree $k$ , so:

\sum_{v\in V}\deg(v)=nk.

Combining with the handshaking lemma gives the key formula:

nk = 2|E|\quad\Longrightarrow\quad |E|=\frac{nk}{2}.

This is stated as the “degree sum formula” for $k$ -regular graphs.

In addition, for such a graph to exist (in the undirected simple-graph setting), $nk$ must be even so that $|E|$ is an integer.

Handshaking lemma - Defines the handshaking lemma: sum of vertex degrees equals twice the number of edges. ↩
Regular graph - States that a $k$ -regular graph on $n$ vertices has $\frac{nk}{2}$ edges and discusses the parity condition for existence. ↩ ↩²

Compute edges for a 46-regular graph with 8 vertices

1
Step 1
Degree $k=46$ and number of vertices $n=8$ .
2
Step 2
Use $|E|=\dfrac{nk}{2}$ , derived from the degree sum / handshaking lemma.
3
Step 3
$|E|=\dfrac{8\cdot 46}{2}=\dfrac{368}{2}=184$ .
4
Step 4
Therefore the correct choice is (iii) $184$ .

Pro Tip

For any $k$ -regular graph on $n$ vertices, edges are always $|E|=\frac{nk}{2}$ . You don’t need the graph structure—only $n$ and $k$ .

Common Pitfall

Don’t use $\frac{n(n-1)}{2}$ (that’s for complete graphs). Here we use regularity + handshaking: $|E|=\frac{nk}{2}$ .

How to Solve Regular-Graph Edge Problems

Read degrees

1) Recognize regularity

Confirm every vertex has the same degree $k$ ."

Apply handshaking

2) Use degree sum

Sum degrees: $nk = 2|E|$ ."

Solve for $|E|$

3) Compute edges

$|E|=\frac{nk}{2}$ ."

Match answer

4) Select option

Compare computed $|E|$ to choices."

Quick Checks & Edge Cases

Given options vs computed number of edges

Computed value: $184$

Knowledge Check

Question 1 of 3

Q1Single choice

A $k$ -regular graph on $n$ vertices has how many edges?

$|E|=\dfrac{nk}{2}$

$|E|=nk$

$|E|=\dfrac{n(n-1)}{2}$

$|E|=\dfrac{n}{2}$

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