Generation of strongly regular graphs from quaternary complex Hadamard matrices

Abstract: A strongly regular graph with parameters ) , , , ( μ λ k v is a regular graph G with v vertices and k degree in which every two adjacent vertices have λ common neighbors and every two non-adjacent vertices have μ common neighbors. In this paper, we propose an algorithm which can be used to construct strongly regular graphs using quaternary complex Hadamard matrices. The order of the strongly regular graph generated by a quaternary complex Hadamard matrix of order n is 2 n . The proposed algorithm has been illustrated by generating a strongly regular graph of order 4 using quaternary complex Hadamard matrix of order 2. Further, higher order strongly regular graphs were tested using Java program. This algorithm could be used to construct strongly regular graphs of order 22n; n∈Z^+.


INTRODUCTION
Let n be even. A quaternary complex Hadamard matrix (Horadam, 2007) of order n is an n n × matrix H with entries from { 1, i} such that ( ) Hadamard matrices can be constructed using different techniques. One such technique is the tensor product or Kronecker product of Hadamard matrices. is the square matrix of order mn defined by (Again, 1985) Let H be a Hadamard matrix of order n . Then the partitioned matrix

Tensor product
is a Hadamard matrix of order n 2 obtained by Sylvester's construction (Wallis, 1975).

Simple graph
A graph G which has no loops and multi-edges is called a simple graph (West, 2000).A simple graph in which each vertex has the same degree k is named as k-regular graph (Diestel, 2005).
is said to be strongly regular graph (Jayathilake et al, 2013) if there are integers λ and µ such that every two adjacent vertices have λ common neighbors and every two nonadjacent vertices have µ common neighbors which is

The adjacency matrix
The adjacency matrix of a simple graph G of order n is an , where I denotes the identity matrix and J denotes the matrix whose all entries are 1 (Chartrand, 1985).

Latin square
A Latin square of order n is an n n × array containing n different symbols such that each symbol occurs exactly once in each row and each column (Harris et al, 2000).
, then L is said to be Symmetric Latin square (van Lint, 2001). In our work Cyclic shifting method (Nishadi et al, 2017) is used to construct symmetric Latin squares.
In this paper, we have proposed an algorithm which can be used to generate strongly regular graphs from quaternary complex Hadamard matrices of order 2 n for 1 ≥ n .

METHODOLOGY
The method of formulating a srg graph from quaternary complex Hadamard matrices is given as a recursive algorithm.

Steps of the proposed algorithm
Consider the quaternary complex Hadamard matrix of order 2 and label columns as column vectors 2 1 , c c . C 2 = Multiply each column vector by the transpose of its conjugate and label the resulting matrices as i C . Applying it to the first column vector and second column vector 2 c , the following block matrices can be obtained. Construction of quaternary complex Hadamard matrix of order 4, C 4 , from C 2 using Sylvester's construction is given below: C 4 = = Then using the above construction ' i C s can be formulated as follows: By using cyclic shifting method, the following Symmetric Latin square can be obtained.
The resulting adjacency matrix of strongly regular graph of order 16=4 2 as follows.  Nishadi et al. We can obtain the strongly regular graphs by applying the same procedure for the quaternary complex Hadamard matrix of order 2 n for 1 ≥ n obtained from Sylvester's construction.

RESULTS AND DISCUSSION
It can be seen that each block that are constructed by the operation ). Every undirected graph has symmetric property. Above method is used to construct strongly regular graph from the Quaternary complex Hadamard matrices of order 4 , 2 = m . A Computer program has been developed by using proposed algorithm, to obtained Strong regular graphs ) , , , ( µ λ k v with large number of vertices. Java Programming and C+ language were used to write the above programme and the following figures were obtained. Using the quaternary complex Hadamard matrix of order 2, we can obtain srg(4,1,0,0)( Figure 2). This graph contains 4 vertices with only one degree and every two adjacent vertices and two non-adjacent vertices have not any common neighbours. Using the quaternary complex Hadamard matrix of order 8, we can obtain srg (64,28,12,12)( Figure 4).