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On strongly regular graphs of order n = 7(2p+1) where 2p+1 is a prime number

In recent years, determining the parameter of strongly regular graph is a important problem of the theory graph. However, when the number of vertices of a strongly regular graph (called n) is any number, determining its parameter is very difficult and nobody can solve this problem. | JOURNAL OF SCIENCE OF HNUE Natural Sci. 2010 Vol. 55 No. 6 pp. 113-127 ON STRONGLY REGULAR GRAPHS OF ORDER n 7 2p 1 WHERE 2p 1 IS A PRIME NUMBER Vu Dinh Hoa Hanoi National University of Education Do Minh Tuan Nam Dinh Teacher Training College E-mail hoavd@fpt.com.vn Abstract.We say that a regular graph G of order n and degree r 1 which is not the complete graph is strongly regular if there exists non-negative integers τ and θ such that Si Sj τ for any two adjacent vertices i and j and Si Sj θ for any two distinct non-adjacent vertices i and j where Sk denotes the neighbourhood of the vertex k. We describe here the parameters n r τ and θ for strongly regular graphs of order 7 2p 1 where 2p 1 is a prime number. Keywords Strongly regular graph conference graph integral graph. 1. Introduction 1.1. Some results of strongly regular graphs In recent years determining the parameter of strongly regular graph is a im- portant problem of the theory graph. However when the number of vertices of a strongly regular graph called n is any number determining its parameter is very difficult and nobody can solve this problem. Therefore we have some results in special cases of n Due to Theorem 1.1 we have recently obtained the following result 2 i there is no strongly regular graph of order 4p 3 if 4p 3 is a prime number. ii the only strongly regular graph of order 4p 1 are conference graphs if 4p 1 is a prime number. Besides 2-5 we have decribed the parameters n r τ and θ for strongly regular graphs of order 2 2p 1 3 2p 1 4 2p 1 5 2p 1 and 6 2p 1 where 2p 1 is the prime number. In this article We now proceed to establish the parameters of strongly regular graphs of order 7 2p 1 where 2p 1 is a prime number as follows. In order to solve the problem. First we have some results about relations of the parameters of strongly regular graphs 113 Vu Dinh Hoa and Do Minh Tuan i A strongly regular graph G of order 4n 1 and degree r 2n with τ n 1 and θ n is called the conference graph. ii A .