Hadamard Logarithmic Series and Inequalities on The parameters of a Strongly Regular Graph

Let G be a primitive strongly regular graph of order n and A its adjacency matrix. In this paper, we first associate an Euclidean Jordan algebra V to G considering the real Euclidean Jordan algebra spanned by the identity of order n and the natural powers of A. Next, by the analysis of the spectra of an Hadamard logarithmic series of V we establish new admissibility conditions on the parameters of the strongly regular graph G.

In this paper we deduce admissibility conditions on the parameters of a strongly regular graph, see [10], in the environment of Euclidean Jordan algebras.
The organization of the paper is as follows. In next section, we present the more relevant definitions and results about Euclidean Jordan algebras necessary for a clear understanding of this paper. In the third section, we present some definitions and results about strongly regular graphs. Finally, in the last section we establish some admissibility conditions over the spectra and over the parameters of a strongly regular graph.

SOME NOTIONS ON EUCLIDEAN JORDAN ALGEBRAS
One finds a clear and simple exposition about Euclidean Jordan algebras on the publication of Farid Alizahed [14]. More detailed surveys about Euclidean Jordan algebras can be encountered in the book Analysis on symmetric cones of Faraut and Korányi [15] and in the book Structure and Representations of Jordan Algebras of Nathan Jacobson [16]. But, we must also say that the monograph of McCrimmon, Taste of Jordan algebras [17] is also a good textbook about Euclidean Jordan Algebras.
Let be a n-dimensional vector space over a field with a bilinear map from to . is a Jordan algebra if and , where .

REMARK 1.
We must say herein that if is an associative algebra over a field with characteristic not equal to 2 and with the operation of multiplication of x by y denoted by xy , that from now on we will denote by xy then we can obtain a Jordan algebra considering on a new operation • defined in the following way ( ) 2 x • y = xy + yx Indeed, we have ( ) 2 ( ) 2 y x yx xy xy yx x y • = + = + = • and we have that: x x y x x y • • = • • and therefore we conclude that is a Jordan algebra when equipped with the product . REMARK 2.We will suppose, from now on that when we say let be a Jordan algebra we mean that is a real finite dimensional Jordan algebra and has a unit element denoted by e.
Let be a Jordan algebra. Then is power associative, this is an algebra such that for any x in the algebra spanned by x and e is associative. Then, there exist real scalars and such that: Where 0 is the zero vector of . Taking in account (1) we conclude that the polynomial p such that is the minimal polynomial of x. When x is not regular the minimal polynomial of x has a degree less that r. The roots of the minimal polynomial of x are the eigenvalues of x.
A real Euclidean Jordan algebra is a Jordan algebra with an inner product , < − − > such that , and if it can not be written as a sum of two nonzero orthogonal idempotents. We say that is a Jordan frame if is a complete system of orthogonal idempotents such that each idempotent is primitive.  We also have that: Therefore is a complete orthogonal system of idempotents of the Euclidean Jordan algebra the second spectral decomposition of x is We suppose that we are using the column notation, this is we consider the notation is a Jordan frame of Indeed, let i be a natural number such that 1 ≤ i ≤ n then we have 2 Let and be two natural numbers such that and such that Then we have:

RESULTS ON STRONGLY REGULAR GRAPHS
A graph G non-null and non-complete, is a strongly regular graph if there are integers , k λ and µ such that G is k regular (k>1) and of order n>3 and any two adjacent vertices of G have exactly λ common neighbors and any two non-adjacent vertices have µ common neighbors.
From now on we will say that a graph G is a ( , ; , ) n k λ µ -strongly regular graph if G is a strongly regular graph of order n and have the parameters , . A It is well known (see, for instance, [18]) that the eigenvalues of G are k , θ and τ , where θ and τ are given by θ = ( λ -μ + Now, we present the admissibility conditions introduced by Delsarte, Goethals and Seidel [20], which establish if G is a ( , ; , ) n k λ µ -strongly regular graph then where f θ and f τ are the multiplicities of the eigenvalues θ and τ of G.
A ( , ; , ) n k λ µ -strongly regular graph G is primitive if and only if G is connect and its complement G is also connected. A strongly regular graph that is not primitive is called imprimitive.
Finally, we must present another property of a strongly regular graph that is: a ( , ; , ) n k λ µ -strongly regular graph is a imprimitive strongly regular graph if and only if 0 µ = or k µ = . .

INEQUALITIES ON THE PARAMETERS OF A STRONGLY REGULAR GRAPH
Herein, we present some admissibility conditions on the spectra and on the parameters of a primitive strongly regular graph but obtained on na asymptotic algebraic way .
Let i,j be natural numbers such that 1 , 3 i j ≤ ≤ and . i j ≠ So, since the idempotents E i and E j are orthogonal relatively to the Jordan product of matrices, then they are orthogonal relatively to the inner product , Now, we consider some notation for defining the Hadamard product of two matrices. We denote the space of real square matrices of order n by ( ) M n ℝ and we consider the Hadamard product and the Kronecker product of two matrices of order , n E and F of ( ) M n ℝ defined in the following way : λ µ > Now, we will analyze the spectra of an Hadamard series associated to the matrix 2 2 | | 2 where A is the adjacency matrix of G . From now on, we will denote the matrix X by x. So, let's consider the Hadamard series Now, we consider the notation: 3 is the spectral decomposition of S x respectively to the Jordan In the following text, we will explain that the eigenvalues q ix s of S x are positive. Let consider the following notation: Since for any two real matrices E and F of order n we have is a Jordan frame of the Euclidean Jordan algebra A that is a basis of A and this Euclidean Jordan algebra is closed for the Hadamard product then we conclude that the eigenvalues of S nx are all positive. Now, we must note that lim , Therefore we have the following expressions for the ix q 's: By an asymptotical analysis of the eigenvalues 3 1 q x we establish the inequality (5) and of Proposition