Farey triangle graph of order N is constructed from Farey Sequence. Consider X and Y axes with vertices as Farey Sequnence in. The Farey triangle graph of order N is formed from Farey triangle graph of order, iteratively as Farey sequence. In this graph, vertices from X to Y axis is joined only if the vertices (or Farey fractions) are same to obtain a Farey triangle.

The Farey triangle graph of order 1 begin with vertices and in both axes. The vertex is the ori-

gin of the Farey triangle graph of order 1. In this graph join the vertices when X and Y axes have the same fractions to obtain a Farey triangle. is constructed from. In this graph the vertices are inserted by the method of the mediant between each pair of consecutive fractions in both axes of. Si-

milarly, we follow the same method to obtain from. The Farey triangle graph of order N can be constructed from Farey triangle graph by using the Farey sum operation denoted as. In the vertices in X and Y axes are irreducible fractions in Farey sequence of order. For each edge in introduced in iteration, could be obtained

from by calculating the mediant between each pair of consecutive fractions in both axes of. Some illustrations are presented below: Figures 1-4 denote the Farey triangle graph of different order, from this graph we define Farey triangle matrix.

In the above illustrations, the like coloured lines denote the edges inserted in successive iterations.

Let be a Farey triangle. The vertices of the triangle are clearly Farey fractions. Let the abscissa be and ordinate, we obtain

The Farey triangle graph forms a matrix [6] ,

where a and b are the numerator of Farey fractions and c denote the order of the Farey sequence. k denote the number of vertices inserted to move from to.

1) Farey triangle matrix of order 2.

2) Farey triangle matrix of order 3

Farey triangle graph of order 3 is derived from Farey triangle graph of order 2. Here two vertices are inserted, so two Farey triangle matrices are constructed.

The sum of the determinants of the Farey triangle matrices of prime order is given by

Proof:

In Farey triangle graph of prime order, vertices are introduced.

The ordinates; are connected only if and it forms a matrix

where

The sum of the determinant of these matrices is

The sum of Farey triangle matrices of prime order is.

Proof:

The Farey triangle matrices of prime order

where.

The complementary Farey triangle graph of order N is a Farey triangle with edges as the line joining vertices whose numerators are complementary with respect to the order of the graph and it forms complementary Farey triangle.

The complementary Farey triangle graph of order can be constructed from complementary Farey triangle graph. The vertices are inserted as in Farey triangle graph. In this graph the

vertices are connected if the sum of the numerators of the fractions in each vertices of X and Y axis is equal to the order of the complementary Farey triangle graph. In complementary Farey triangle graph of order 2, we be-

gin with vertices and in both axes. The vertex is the origin of the complementary Farey

triangle graph. In this graph the vertices are inserted by the method of the mediant between each pair of consec-

utive fractions in both axes of. We follow the same method to obtain from.

Figures 5-7 denotes the complementary Farey Triangle Graph of different orders, from this graph we define Complementary Farey Triangle Matrix. Some illustrations are presented below:

The vertices of the complementary Farey triangle namely Farey fractions are used to construct this matrix. Let the abscissa be and ordinate. Join these vertices to form a complementary Farey triangle graph and correspondingly complementary Farey triangle matrix as given below.

where a and b are the numerator of Farey fractions and c denote the order of the Farey sequence. k denote the number of vertices inserted to move from to.

The sum of determinants of the complementary Farey triangle matrices of prime order p is

Proof:

Consider the complementary Farey triangle matrices of prime order.

;

where