# Complement of Graph in Discrete mathematics

In discrete mathematics, the simple graph is indicated as G, and the Complement of this graph is indicated as G`. If G` is the complement of a simple graph G, then it must contain the following things:

- The complement graph must contain all the vertices of graph G.
- If there are two vertices v and w and the original graph G does not contain any edge between these vertices, in this case, the complement graph must contain an edge between these two vertices v and w.

The simple example of complement of the graph is described as follows:

In this example, we have a simple graph on the left side, which is described as G and its complement graph on the right side, which is described as G`.

### Relation between G and G`:

A simple graph G and complement graph G` contains some relations, which are described as follows:

1. The number of vertices in graph G equals to the number of vertices in its complement graph G1`. The symbolic representation of this relation is described as follows:

2. The sum of total number of edges in both the graphs, i.e., simple graph G and its complement graph G` will equal to the total number of edges in a complete graph. The symbolic representation of this relation is described as follows:

Here n is used to indicate the total number of vertices in the graph

**Important Points:**

When we are learning the term complement of a graph, then we should know some points, which are described as follows:

- Size of the graph will be equal to the total number of vertices in the graph
- Same as the Order of graph will be equal to the number of vertices in the graph

### Examples of Complement of Graph

There are various examples of Complement graphs, and some of them are described as follows:

**Example 1:** In this example, we have a simple graph G, which contains 21 edges and 10 vertices. Now we will find out the number of edges in the graph G`.

**Solution:** From the question, we have the following details about graph G:

Number of vertices n = 10

Number of edges |E(G)| = 21

As we know that |E(G)| + |E(G`)| = n(n-1) /2

Now we will put the values of n and |E(G)| in this formula, and get the following details:

21 + |E(G`)| = 10*(10-1) /2

| E(G`)| = 45-21

|E(G`)| = 24

**Example 2:** In this example, we have a simple graph G and a complement graph G`. The graph G contains the 30 edges and G` contains the 36 edges. Now we will find out the number of vertices in graph G.

**Solution:** From the question, we have the following:

In graph G, the number of edges |E(G)| = 30

In graph G`, the number of edges |E(G`)| = 36

As we know that |E(G)| + |E(G`)| = n(n-1) /2

Now we will put the values of |E(G)| and |E(G`)| in this formula, and get the following details:

30+36 = n(n-1) /2

n(n-1) = 132

n^{2} â€“ n â€“ 132 = 0

Now we will solve the above equation with the help of a quadratic equation. The quadratic equation is described as follows:

n = (-b **Â±** âˆšb^{2} â€“ 4ac) / 2a

When the equation is in standard form, then we will find the value of a, b and c from the original equation n^{2} â€“ n â€“ 272 = 0, and plus them into the above explained quadratic formula like this:

n^{2} â€“ n â€“ 132 = 0

a = 1

b = -1

c = -132

n = {-(-1) **Â±** âˆš(-1)^{2} â€“ 4*1*(-132))} / 2*1

n = {1 **Â±** âˆš1 â€“ (-528)} /2

n = {1 **Â±** 23)} /2

n = 24 /2

n = 12

Thus, graph G contains the number of vertices as G = 12.

**Example 3:** In this example, we have a simple graph G, which contains the order n. Here the size of a simple graph G is 56, and the size of its complement graph G` is 80. Now we will find out the value of n.

**Solution:** Here Size of a graph = Number of edges in graph

From the question, we have the following:

In graph G, the number of edges |E(G)| = 56

In graph G`, the number of edges |E(G`)| = 80

As we know that |E(G)| + |E(G`)| = n(n-1) /2

Now we will put the values of |E(G)| and |E(G`)|, and get the following details:

56+80 = n(n-1) /2

n(n-1) = 272

n^{2} â€“ n â€“ 272 = 0

Now we will solve this equation with the help of a quadratic equation. The quadratic equation is described as follows:

n = (-b **Â±** âˆšb^{2} â€“ 4ac) / 2a

When the equation is in standard form, then we will find the value of a, b and c from the original equation n^{2} â€“ n â€“ 272 = 0, and plus them into the above explained quadratic formula like this:

n^{2} â€“ n â€“ 272 = 0

a = 1

b = -1

c = -272

n = {-(-1) **Â±** âˆš(-1)^{2} â€“ 4*1*(-272))} / 2*1

n = {1 **Â±** âˆš1 + 1088)} /2

n = {1 **Â±** 33}

n = 34 /2

n = 17

Thus, graph G contains the number of vertices as G = 17.

In other words, we can say that the Order of graph G = 17.