---
title: "Define regular graph with example"
subject: "Discrete Structure"
chapter: 4
key: "semester_4/discrete_structure:define-regular-graph-example"
source: https://www.bctengineer.com/q/semester_4/discrete_structure/define-regular-graph-example
site: BCT Engineering
---

# Define regular graph with example

Discrete Structure · Chapter 4: Graph Theory

- Asked in: 2070 Bhadra, 3 marks; 2074 Bhadra, 2 marks; 2070 Magh, 3 marks (asked 3 times)
- Video: [What are Regular Graphs? | Graph Theory](https://www.youtube.com/watch?v=qHLXZCxoUdE) — Wrath of Math
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/discrete_structure/define-regular-graph-example
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-4

## Question

Define regular graph with suitable example.

## Short answer

A regular graph has every vertex with the same degree k, called k-regular. Examples: C5 is 2-regular, K4 is 3-regular, and the cube Q3 is 3-regular.

## Answer

A **regular graph** is a simple graph in which every vertex has the same degree. If every vertex has degree k, it is called **k-regular**.

Properties:
- A k-regular graph on n vertices has nk/2 edges (by the handshaking theorem), so n·k must be even.
- A 0-regular graph has no edges, a 1-regular graph is a perfect matching, and a 2-regular graph is a disjoint union of cycles.

Examples:
- **Cₙ** (cycle) is 2-regular; C₄ is the square.
- **Kₙ** (complete graph) is (n − 1)-regular; K₄ is 3-regular with 6 edges.
- **Q₃** (cube) is 3-regular with 8 vertices and 12 edges.
- The Petersen graph is 3-regular on 10 vertices.

```text
  a ------ b          every vertex has degree 2
  |        |          (2-regular, n = 4, edges = 4·2/2 = 4)
  d ------ c
```
This is C₄. In contrast, the path P₃ (a–b–c) is not regular, since the end vertices have degree 1 and the middle one has degree 2.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. A regular graph is one in which:
   A. Every vertex has the same degree
   B. Every pair of vertices is adjacent
   C. Every vertex has degree 2
   D. The graph is a tree
   Answer: A. k-regular means every vertex has degree k. (Key point: Definition)

2. A k-regular graph on n vertices has how many edges?
   A. nk
   B. nk/2
   C. n+k
   D. n(n-1)/2 always
   Answer: B. By the handshaking theorem the degree sum nk equals 2e, so e = nk/2 and nk must be even. (Key point: Edge count)

3. Which graph is NOT regular?
   A. Cycle C4
   B. Complete graph K4
   C. Path P3 (a-b-c)
   D. Cube Q3
   Answer: C. In P3 the end vertices have degree 1 and the middle has degree 2. (Key point: Example)