## Complement of Graph in Discrete mathematicsIn 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
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 GraphThere are various examples of Complement graphs, and some of them are described as follows:
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
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 Now we will solve the above equation with the help of a quadratic equation. The quadratic equation is described as follows: n = (-b When the equation is in standard form, then we will find the value of a, b and c from the original equation n n a = 1 b = -1 c = -132 n = {-(-1) n = {1 n = {1 n = 24 /2 n = 12 Thus, graph G contains the number of vertices as G = 12.
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 Now we will solve this equation with the help of a quadratic equation. The quadratic equation is described as follows: n = (-b When the equation is in standard form, then we will find the value of a, b and c from the original equation n n a = 1 b = -1 c = -272 n = {-(-1) n = {1 n = {1 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. |

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