A Study on Permanent of L-R Hexagonal Fuzzy Number Matrices

In this paper, it is shown that a number of properties of permanent of both square and non-square matrices present under fuzzy environment. We establish the basic formulas of determinant and permanent of matrices which contains the new properties to compare the both notions. We investigate the permanent of square L-R hexagonal fuzzy matrix (L-R HFM) using in different ways from partial derivatives. The notions of permanent of nonsquare L-R hexagonal fuzzy matrix are defined. Moreover, we derive some of the standard properties and constant matrix with the aid of above notion.


Introduction
The permanent has a rich structure when restricted to certain class of matrices, particularly, matrices of zeros and ones (entrywise) non-negative matrices and positive semi defined matrices.Furthermore, there is a certain similarity of its properties over the class of non-negative matrices and class of positive semi defined matrices.Romanwicz and Grabowski [6] used permanent of square matrix.Permanent is also used in graphtheoretic interpretations.One is, as the sum of the weights of cycle covers of a directed graph, yet another one is as the sum of weights of perfect matching in a bipartite graph.Definition 2.1.(L-R Hexagonal Fuzzy Number) [10] An L-R hexagonal fuzzy number denoted by  An L-R hexagonal fuzzy number is said to be symmetric, if the sum of both its spreads are equal, i.e., if and it is denoted by LR n m α α

Arithmetic operations on L-R hexagonal fuzzy numbers (HFNs)
Here we introduce the definition of arithmetic operations between two L-R hexagonal fuzzy numbers (L-R HFNs) are given below. ).

(Ranking Function)
We define a ranking function which maps each fuzzy numbers to real line ( ) represented the set of all hexagonal fuzzy numbers.If R be any linear ranking functions, then ( )

L-R Hexagonal Fuzzy Matrices (L-R HFMs)
In this section, we propose new definitions of L-R hexagonal fuzzy number matrix and its corresponding matrix operations.( ),  ), where k is scalar.
We now define some special types of L-R HFMs corresponding to special classical matrices.

Definition 3.2. (Zero L-R Hexagonal Fuzzy Matrix)
An L-R hexagonal fuzzy matrix (L-R HFM) is said to be a zero L-R HFM if all its

A Comparison between Permanent and Determinant of Matrices
In this section, we have to compare both the permanent and determinant of matrices using crisp matrix and find out the new results are justified.
where n s denotes the symmetric group of order n.
The definition of permanent [4] is similar to the definition of determinant except the sign of each term in summation.The number of terms over summation are both cases but the sign associated in each term are all positive in case of permanent.The permanent cannot compete with determinant, in terms of the depth of theory and breadth of applications, but it is safe to say that the permanent also exhibits both these characteristic in ample measure, a fact that has not receive enough attention.
Here three ways to calculate ( ) The classical formula using all the permutation in 3 S is ( ) Compare with all the three formulas, we take the first formula to apply all problems throughout this paper and following an example is, Let us consider an example to illustrate both the determinant and permanent of crisp matrix.
is the positive square matrix of order n, then ( ) ( ).
is even square matrix of order n, then ( ) ( ).
is odd square matrix of order n, then ( ) ( ).
det B B per ≤ 3. If any one of the row (or) column of a crisp matrix of order n is negative, then the permanent of a matrix is negative.
4. If any one of the row (or) column of a crisp matrix of order n is negative, then

Verifications:
The above said special properties of permanent have been verified by the counter examples.
Therefore, the permanent of a matrix is negative.

( ) ( ).
det A A per < In particular, matrix will become zero matrix and triangular matrix, then

Permanent of Square L-R Hexagonal Fuzzy Matrices
In this section, we investigate the permanent of L-R hexagonal fuzzy matrix involving partial derivatives of homogeneous polynomial of degree n and characteristic p over a fuzzy field .F

Permanent with partial derivatives
Let LR A ˆ be an n n × L-R hexagonal fuzzy matrix defined as ) we can write ( ) where the summation is over all sequence ( ) for any k and l are defined by differentiation  ( ).
Again by differentiation and recalling equation (5.2), we have

Permanent of L-R hexagonal fuzzy number from partial derivatives
To obtain a non-zero monomial in these variables from the operator that acts upon f.Now, we consider the monomial in f are of the form where k is a homogeneous polynomial of degree n.Since since the even cycles can be neglected when we calculate the length of permutation mod 2. Also, q is an even permutation if and only if r is even.When k is produced by the above partial differentiation from , ˆLR Z there is a factor of ( ) in general function of p.In addition each of these ways of producing k gives the same sign as q, since the 1 − p permutations from the determinants have a product that is q and so the signs multiply to give ( ) ( ) .
1 sgn r q − = Hence the total coefficient of k in the partial derivative of LR A ˆ and hence of ( ) Thus the formula in the theorem is true.From the corollary, Hence the proof.

Permanent of Non-Square L-R Hexagonal Fuzzy Matrices
In this section, we introduce the notion of permanent of L-R HFM to define the order n m × .Also, we define the notion of constant L-R HFM in the same order and its relevant properties are discussed.)   ∑ ∈ β β α α = S q q q q q q q n m

A 2 β
Here the points of m and n, with membership value of 1, is called the flat region of mean value are the four distinct left and right spreads of , ~hLR A respectively.
-R HFMs of same order.Then we have the following:1.

s
denotes the symmetric group of all permutations of the indices { } for even permutations and −1 for odd permutations.matrix of order , n n × then the permanent of A is denoted by 1) and (4.2),

1 ⋯A
Study on Permanent of L-R Hexagonal Fuzzy Number Matrices Earthline J. Math.Sci.Vol. 2 No. 1 (2019), -R HFM over a fuzzy field F of characteristic p. Then it is defined as n × L-R HFM over a fuzzy field of characteristic p. Then

c
that k must be a similar form corresponding to the unique permutation .cycle of size is fixed point of one.Then we note that -R HFM over a fuzzy field F characteristic three.Then

Property 6 . 4 .
If any two rows (or columns) of a non-square L-R HFM LR A ˆ are interchanged, then the permanent value remains unchanged.non-square L-R HFM obtain from LR A ˆ by interchanging the rth and sth rows ( )

Property 6 . 5 .
Let LR A ˆ be a non-square L-R HFM of order .n m × If a row is multiplied by scalar k, then the permanent value is k non-square L-R HFM obtained by multiplying k to a

. Special properties of permanent
Properties of permanent are also presented below.
Earthline J. Math.Sci.Vol. 2 No. 1 (2019), 39-67 47 1.For a crisp matrix This is because a cycle of size one in q corresponds to an ′ ~ of k in each non-trivial cycle must be assigned to one of the A Study on Permanent of L-R Hexagonal Fuzzy Number Matrices Earthline J. Math.Sci.Vol. 2 No. 1 (2019), 39-67