Define the following types of graphs with suitable examples and properties:…

Types of Graphs and Their Properties

(a) Cycle Graph (Cn​)
A cycle graph is a closed path in which each vertex is connected to exactly two vertices.
Property: It contains n vertices and n edges, and every vertex has degree 2 (2-regular graph).

(b) Wheel Graph (Wn​)
It is formed by joining a central hub vertex to all vertices of a cycle Cn−1​.
Property: Total edges = 2(n−1); hub degree = n − 1, outer vertices have degree 3.

(c) Regular Graph
A graph is called k-regular if every vertex has degree k.
Property: Cn​ is 2-regular and Kn​ is (n−1)-regular.

(d) Complement of a Graph (G')
A graph on the same vertex set where edges exist only where they are absent in G.
Property: ∣EG'∣ = n(n−1) / 2−∣EG∣|.

Relationship & Illustration

• G ∪ G' = Kn​ (Complete Graph)

• degG(v) + degG'(v) = n−1

• ∣EG∣+∣EG'∣ = n (n−1) / 2

Conclusion:
These graphs highlight structural patterns, while complement graphs provide a dual perspective, essential for analyzing graph properties in depth.