Discrete Math - Simple Graph Types

Table of Contents

In previous lessons, you learned about graph models and some basic graph terminologies.

In this lesson, you will learn about simple graph types, we learned earlier that a simple graph is one in which each edge has two unique vertices.

Simple Graph Types

The following is a list of simple graph types that we are going to explore.

Complete Graph
Cycles
Wheels
n-Cube
Bipartite Graph
Complete Bipartite Graph

Let us discuss each one them.

Complete Graph

A complete graph on n vertices, denoted by $K_n$ is a simple graph that contains exactly one edge between each pair of distinct vertices. It any edge from the pair of distinct vertices is not connected then it is called non-complete.

Here are some examples of complete graph.

Cycles

A cycle is denoted as $C_n$ where $n \geq 3$ consists of n vertices v₁, v₂, …, v_nand edges { v₁, v₂ }, { v₂, v₃ }, …, { v_n, v₁}.

Here are some examples of cycles.

Wheels

A wheel graph is easy to construct. You get a wheel, $W_n$ when you add an additional vertex to a cycle $C_n$ , for $n \geq 3$ and place it in center of the cycle. This will connect each of the n vertices of cycle using new edges.

Some examples of wheels.

n-Cubes

A n-dimensional hyper-cube or n-cube is denoted by $Q_n$ is simple graph that has vertices representing the $2^n$ bit-strings of length n.

In this scheme, two vertices are adjacent iff the bit strings differ only in one bit position. If this is confusing for you, see the following example.

Suppose $Q_1$ is given by following diagram.

The n-Cube, $Q1$ graph contains bit string length of 1. To construct $Q2$ from $Q1$ write another copy of $Q1$ and prefix 0 and 1 to each opposite end.

Similarly, to draw $Q3$ graph from $Q2$ , do same as above, make a copy of $Q2$ and add 0 and 1 to opposite ends everywhere. Then join them by edges.

Bipartite Graphs

There is a special type of simple graph called bipartite graph. A simple graph $G$ is bipartite if its vertex set $V$ can be partitioned into two disjoint sets – $V1$ and $V2$ , such that every edge in the graph $G$ connects a vertex in $V1$ and $V2$ . It means no edge in $G$ connects two distinct vertices in either $V1$ or $V2$ .

When this condition is holds, then you call the pair $(V1, V2)$ bi-partition of $V$ of graph $G$ .

Consider following graphs.

In the first graph, the vertex set $V$ is divided into

$V1 = { S, R }$ $V2 = { A, B, C }$

This graph is a bipartite because there is no edge between nodes of same set. Every edge is connecting two vertices – one from each subset of $V$ .

The second graph, the vertex set is divided into

$V1 = { A, B }$ $V2 = { C, D }$

This graph is not a bipartite because $A$ and $B$ has an edge and both $A$ and $B$ belong to subset $V1$ .

Complete Bipartite Graph

If you recall, a complete graph is a graph $K_n$ . However, a complete bipartite graph $K_{m,n}$ has its vertex set $V$ partitioned into m and n vertices, respectively with an edge between two vertices if and only if one vertex is in the first subset and the other vertex is in the second subset.

The total number of vertices in the set $V = m + n$ and the total number of edges is $E = m * n$ in a complete bipartite graph.