Regular Graph in Graph Theory

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:

  1. 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.

  2. 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.

  3. Cycle(Cn) is always 2 Regular.
    Proof:
    In Cycle (Cn) each vertex has two neighbors. So, they are 2 Regular.

  4. 2 Regular graphs consists of Disjoint union of cycles and Infinite Chains.

  5. 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

  6. A K-dimensional Hyper cube (Qk) is a K Regular graph.
    Below is a 3-dimensional Hyper cube(Q3) which is a 3 Regular graph.



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