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In this paper, we introduce Farey triangle graph , Farey triangle matrix , complementary Farey triangle graph and complementary Farey triangle matrix , and we derive some properties of the following matrices.

A Farey sequence of order N is a set of irreducible fractions between 0 and 1 arranged in an increasing order, the denominators of which do not exceed N.

angle graph

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 1 begin with vertices

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.

milarly, we follow the same method to obtain

from

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

Let

The Farey triangle graph forms a matrix [

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

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

The ordinates

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

The complementary Farey triangle graph

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

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

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

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

1) Complementary Farey triangle matrix of order 2.

2) Complementary Farey triangle matrix of order 3

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

A.Gnanam,C.Dinesh, (2015) Farey Triangle Graphs and Farey Triangle Matrices. Advances in Pure Mathematics,05,738-744. doi: 10.4236/apm.2015.512067