Prerequisite: Graph Theory Basics – Set 1, Set 2

**Regular Graph:**

A graph is called regular graph if degree of each vertex is equal. A graph is called **K regular** if degree of each vertex in the graph is K.

**Example:**

Consider the graph below:

Degree of each vertices of this graph is 2. So, the graph is *2 Regular*. Similarly, below graphs are *3 Regular* and *4 Regular* respectively.

**Properties of Regular Graphs:**

- A complete graph N vertices is
*(N-1)*regular.

__Proof__:

In a complete graph of N vertices, each vertex is connected to all (N-1) remaining vertices. So, degree of each vertex is (N-1). So the graph is (N-1) Regular. - For a K Regular graph, if K is odd, then the number of vertices of the graph must be even.

__Proof__:

Lets assume, number of vertices, N is odd.

From Handshaking Theorem we know,

Sum of degree of all the vertices = 2 * Number of edges of the graph …….(1)

The R.H.S of the equation (1) is a even number.For a K regular graph, each vertex is of degree K. Sum of degree of all the vertices = K * N, where K and N both are odd.So their product (sum of degree of all the vertices) must be odd. This makes L.H.S of the equation (1) is a odd number. So L.H.S

*not equals*R.H.S. So our initial assumption that N is odd, was wrong. So, number of vertices(N) must be even. - Cycle(C
_{n}) is always 2 Regular.

__Proof__:

In Cycle (C_{n}) each vertex has two neighbors. So, they are 2 Regular. - 2 Regular graphs consists of
*Disjoint union of cycles*and*Infinite Chains*. - Number of edges of a K Regular graph with N vertices = (N*K)/2.

__Proof__:

Let, the number of edges of a K Regular graph with N vertices be E.

From Handshaking Theorem we know,Sum of degree of all the vertices = 2 * E

N * K = 2 * E

or, E = (N*K)/2 - A K-dimensional Hyper cube (Q
_{k}) is a K Regular graph.

Below is a 3-dimensional Hyper cube(Q_{3}) which is a 3 Regular graph.