Euclidean Jordan algebras and some conditions over the spectra of a strongly regular graph

Let G be a primitive strongly regular graph G such that the regularity is less than half of the order of G and A its matrix of adjacency, and let A be the real Euclidean Jordan algebra of real symmetric matrices of order n spanned by the identity matrix of order n and the natural powers of A with the usual Jordan product of two symmetric matrices of order n and with the inner product of two matrices being the trace of their Jordan product. Next the spectra of two Hadamard series of A associated to A is analysed to establish some conditions over the spectra and over the parameters of G.

A good exposition about Jordan algebras can be founded in the beautiful work of K. McCrimmon, A taste of Jordan algebras, see [25], or for a more abstract survey one must cite the work of N. Jacobson, Structure and Representations of Jordan Algebras, see [26] and the PhD thesis of Michael Baes, Spectral Functions and Smoothing Techniques on Jordan Algebras, see [27].
For a well based understanding of the results of Euclidean Jordan algebras we must cite the works of Faraut and Korányi, Analysis on Symmetric Cones, see [28], the PhD thesis Jordan algebraic approach to symmetric optimization of Manuel Vieira, see [29], and the PhD thesis A Gershgorin type theorem, special inequalities and simultaneous stability in Euclidean Jordan algebras of Melanie Moldovan, see [30].
But for a very readable text on Euclidean Jordan algebras we couldn't avoid of indicating, the chapter written by F. Alizadeh and S. H. Schmieta, Symmetric Cones, Potential Reduction Methods and Word-By-Word Extensions of the book Handbook of semi-definite programming, Theory, Algorithms and Applications, see [31], and the chapter, written by F. Alizadeh, An Introduction to Formally Real Jordan Algebras and Their Applications in Optimization of the book Handbook on Semidefinite, Conic and Polynomial Optimization, see [32].
In this paper we establish some admissibility conditions in an algebraic asymptotically way over the parameters and over the spectra of a primitive strongly regular graph. This paper is organized as follows. In the second section we expose the most important concepts and results about Jordan algebras and Euclidean Jordan algebras, without presenting any proof of these results. Nevertheless some bibliography is present on the subject. In the following section we present some concepts about simple graphs and namely strongly regular graphs needed for a clear understanding of this work. In the last section we consider a three dimensional real Euclidean Jordan algebra A associated to the adjacency matrix of a primitive strongly regular graph and we establish some admissibility conditions over, in an algebraic asymptotic way, the spectra and over the parameters of a strongly regular graph.

Principal results on Euclidean Jordan algebras
Herein, we describe the principal definitions, results and the more relevant theorems of the theory of Euclidean Jordan algebras without presenting the proof of them.
To make this exposition about Euclidean Jordan algebras we have recurred to the the monograph, Analysis on Symmetric Cones of Faraut and Kóranyi [28], and to the book A taste of Jordan algebra of Kevin McCrimmon [25]. But for general Jordan algebras very readable expositions can be found in the book Statistical Applications of Jordan algebras of James D. Malley [17]. Now, we will present only the main results about Euclidean Jordan algebras needed for this paper. A Jordan algebra A over a field K with characteristic 6 ¼ 2 is a vector space over the field K with a operation of multiplication Å such that for any x and y in A, x Å y = y Å x and x Å (x 2Å Å y) = x 2Å Å (x Å y), where x 2Å = x Å x. We will suppose throughout this paper that if A is a Jordan algebra then A has a unit element that we will denote it by e.
When the field K is the field of the reals numbers we call the Jordan algebra a real Jordan algebra. Since we are only interested in finite dimensional real Jordan algebras with unit element, we only consider Jordan algebras that are real finite dimensional Jordan algebras and that have an unit element e and that are equipped with an operation of multiplication that we denote by Å.
The real vector space of real symmetric matrices, A ¼ Sym ðn; RÞ, of order n, with the operation x Å y ¼ xyþyx 2 is a real Jordan algebra.
We must note, that we define the powers of an element x in A in the usual way 2 2 ¼ x 2 and therefore by induction over N we conclude that x kÅ ¼ x k for any natural number k, where x k represents the usual power of order k of a squared symmetric matrix. A is a Jordan algebra since for x and y in A we have So C Å D is a symmetric matrix of M n ðRÞ, but from property iv) we conclude that C Å D ¼ P l¼1 k¼1 c k B k ; therefore C Å D in B: Now, we must say, that for any matrices X and Let A be a n-dimensional Jordan algebra. Then A is power associative, this is, is an algebra such that for any x in A the algebra spanned by x and e is associative. For x in A we define rank (x) as being the least natural number k such that fe; x 1Å ; . . . ; x kÅ g is a linearly dependent set and we write rank (x) = k. Now since dimðAÞ ¼ n then rank ðxÞ n: The rank of A is defined as being the natural number r ¼ rank ðAÞ ¼ max f rank ðxÞ : x 2 Ag: An element x in A is regular if rank ðxÞ ¼ r: Now, we must observe that the set of regular elements of A is a dense set in A. Let's consider a regular element x of A and r = rank (x).
Then, there exist real numbers a 1 ðxÞ; a 2 ðxÞ; . . . ; a rÀ1 ðxÞ and a r ðxÞ such that where 0 is the zero vector of A. Taking into account (1) we conclude that the polynomial p(x, À) define by the equality (2).
is the minimal polynomial of x. When x is a non regular element of A the minimal polynomial of x has a degree less than r. The polynomial p(x, À) is called the characteristic polynomial of x. Now, we must say that the coefficients a i ðxÞ are homogeneous polynomials of degree i on the coordinates of x on a fixed basis of A. Since the set of regular elements of A is a dense set in A then we extend the definition of characteristic polynomial to non regular elements of A by continuity. The roots of the characteristic polynomial p(x, À) of x are called the eigenvalues of x. The coefficient a 1 ðxÞ of the characteristic polynomial p(x, À) is called the trace of x and we denote it by Trace (x) and we call the coefficient a r ðxÞ the determinant of x and we denote it by Det (x). Let A be a real finite dimensional associative algebra with the bilinear map ðx; yÞ7 !x Å y. We introduce on A a structure of Jordan algebra by considering a new product defined by x y ¼ ðx Å y þ y Å xÞ=2 for all x and y in A. The product is called the Jordan product of x by y. Let A be a real Jordan algebra and x be a regular element of A: Then we have rank ðxÞ ¼ r ¼ rank ðAÞ: We define the linear operator L Å ðxÞ such that L Å ðxÞz ¼ x Å z; 8z 2 A: We define the real vector space R½x by R½x ¼ fz 2 A : 9c 0 ; c 1 ; Á Á Á ; c rÀ1 2 R : z ¼ c 0 e þ c 1 x 1Å þ Á Á Á þ c rÀ1 x ðrÀ1ÞÅ g: The restriction of the linear operator L Å ðxÞ to R½x we call L 0 Å ðxÞ: We must note now that trace ðL 0 Å ðxÞÞ ¼ a 1 ðxÞ ¼ Trace ðxÞ and det ðL 0 Å ðxÞÞ ¼ a r ðxÞ ¼ Det ðxÞ: A Jordan algebra is simple if and only if does not contain any nontrivial ideal. An Euclidean Jordan algebra A is a Jordan algebra with an inner product Á|Á such that L Å ðxÞyjz ¼ yjL Å ðxÞz, for all x, y and z in A: Herein, we must say that an Euclidean Jordan algebra is simple if and only if it can't be written as a direct sum of two Euclidean Jordan algebras. But it is already proved that every Euclidean Jordan algebra is a direct orthogonal sum of simple Euclidean Jordan algebras.
The Jordan algebra A ¼ Sym ðn; RÞ equipped with the Jordan product x Å y ¼ xyþyx 2 with xy and yx the usual products of matrices of order n, x by y and of y by x and with the inner product xjy ¼ trace ðL Å ðxÞyÞ for x and y in A is an Euclidean Jordan algebra. Indeed, we have Now, we will show that the A nþ1 ¼ R nþ1 is an Euclidean Jordan algebra relatively to the inner product We have the following calculations Hence, we conclude that L Å ðxÞyjz ¼ yjL Å ðxÞz: And therefore A nþ1 is an Euclidean Jordan algebra. Now, we will analyse the rank of the Euclidean Jordan algebra A ¼ Sym ðn; RÞ. Let consider the element x of A with n distinct eigenvalues k 1 , k 2 ,. . ., k nÀ1 and k n , and B ¼ fv 1 ; v 2 ; Á Á Á ; v n g an orthonormal basis of R n of eigenvectors of x such that xv i ¼ k i v i for i=1,Á Á Á, n. Considering the notation e = I n , then we have: Therefore, the set X nÀ1 ¼ fe; x 1Å ; x 2Å ; Á Á Á ; x nÀ1Å g is a linearly independent set of A if and only if the set is a linearly independent set of R n : But, since then the set S nÀ1 is a linearly independent set of R n and therefore the set is a linearly independent set of A: The set X n ¼ fe; x 1Å ; x 2Å ; Á Á Á ; x nÅ g is a linear dependent set of A since the set is a linearly dependent set of R n because the dimension of R n is n. Therefore, we conclude that rank (x) = n. Let x be an element of A with k distinct non null eigenvalues k i s, and let v i 1 ; v i 2 ; Á Á Á ; v i lÀ1 and v i l be an orthonormal basis of eigenvectors of x of the eigenvector space of x kÅ g is a linearly dependent set of A since dimðR k Þ ¼ k and therefore rank ðxÞ ¼ k: If x has k distinct eigenvalues where kÀ1 eigenvalues are non null and one is null then one proves one a similar way that rank ðxÞ ¼ k. Therefore we conclude that rank ðAÞ ¼ n and the regular elements of A are the elements x of A with n distinct eigenvalues.
The characteristic polynomial of a regular element of A is a monic polynomial of minimal degree n ¼ rank ðAÞ: Now, let x be an element of A with n distinct eigenvalues k 1 , k 2 ,Á Á Á, k nÀ1 and k n then by the Theorem of Cayley-Hamilton we conclude that the polynomial p such that pðkÞ ¼ ðk À k 1 Þðk À k 2 Þ Á Á Á ðk À k n Þ is the monic polynomial of minimal degree of x. Therefore since the monic polynomial of minimal degree of element x is unique we conclude that the characteristic polynomial of x, p(x, À) is such that pðx; kÞ ¼ pðkÞ: This is pðx; kÞ ¼ ðk À k 1 Þðk À k 2 Þ Á Á Á ðk À k n Þ: So, we have pðx; Now, we will show that rank ðA nþ1 Þ ¼ 2 To came to this conclusion, we will firstly show that for Therefore, the set 1 0 and since is a linearly dependent set of A nþ1 then rank x 1 0 ! ¼ 1: And, therefore rank ðA nþ1 Þ ¼ 2: And the regular elements of A nþ1 are the elements of A nþ1 such that x 6 ¼ 0: Since, when x 6 ¼ 0 we have Then, supposing x 6 ¼ 0 and considering the notation ; we conclude that the caractheristic polynomial of x is pðx; kÞ ¼ k 2 À 2x 1 k þ ðx 2 1 À jjxjj 2 Þ: Therefore, the eigenvalues of x are k 1 ðxÞ ¼ x 1 À jjxjj and k 2 ðxÞ ¼ Let A be a real Euclidean Jordan algebra with unit element e.
An element g of A is a primitive idempotent if it is a non null idempotent of A and if cannot be written as a sum of two orthogonal nonzero idempotents of A. We say that fg 1 ; g 2 ; . . . ; g l g is a Jordan frame of A if fg 1 ; g 2 ; . . . ; g l g is a complete system of orthogonal idempotents such that each idempotent is primitive.
Let consider the matrices E jj of the Euclidean Jordan algebra A ¼ Sym ðn; RÞ with j = 1,Á Á Á, n where E jj is the square matrix of order n such that (E jj ) jj = 1 and ( Let consider the Euclidean Jordan algebra A nþ1 and x non zero element of A nþ1 . Then the set is a Jordan frame of the Euclidean Jordan algebra A nþ1 : Indeed, we have: and, we have Therefore we conclude that fg 1 ; g 2 g is a Jordan frame of A nþ1 since rank ðA nþ1 Þ ¼ 2: Theorem 1. ( [28], p. 43). Let A be a real Euclidean Jordan algebra. Then for x in A there exist unique real numbers b 1 ðxÞ; b 2 ðxÞ; . . . ; b k ðxÞ; all distinct, and a unique complete system of orthogonal idempotents fg 1 ; g 2 ; . . . ; g k g such that The numbers b j ðxÞ's of (3) are the eigenvalues of x and the decomposition (3) is the first spectral decomposition of x.
Theorem 2. ( [28], p. 44). Let A be a real Euclidean Jordan algebra with rank ðVÞ ¼ r: Then for each x in A there exists a Jordan frame fg 1 ; g 2 ; Á Á Á ; g r g and real numbers b 1 ðxÞ; Á Á Á ; b rÀ1 ðxÞ and b r ðxÞ such that Remark 1. The decomposition (4) is called the second spectral decomposition of x. And we have relatively to the Jordan frame An Euclidean Jordan algebra is called simple if and only if have only trivial ideals. Any simple Euclidean Jordan algebra is isomorphic to one of the five Euclidean Jordan algebras that we describe below: (i) The spin Euclidean Jordan algebra A nþ1 .
(ii) The Euclidean Jordan algebra A ¼ Sym ðn; RÞ with the Jordan product of matrices and with an inner product of two symmetric matrices as being the trace of their Jordan product. We now describe the Pierce decomposition of an Euclidean Jordan algebra relatively to one of its idempotents. But, first we must say that for any nonzero idempotent g of an Euclidean Jordan algebra A the eigenvalues of the linear operator L Å ðgÞ are 0; 1 2 and 1 and this fact permits us to say that, considering the eigenspaces and Aðg; 1Þ ¼ fx 2 A : L Å ðgÞðxÞ ¼ 1xg of L(g) associated to these eigenvalues we can decompose A as an orthogonal direct sum A ¼ V ðg; 0Þ þ V g; 1 2 À Á þ V ðg; 1Þ: Now, we will describe the Pierce decomposition of the Euclidean Jordan algebra A ¼ Sym ðn; RÞ relatively to an idempotent of the form Now, let calculate A C; 1 2 À Á : So, let consider the following equivalences.
For, other hand if we consider a Jordan frame S ¼ fg 1 ; g 2 ; Á Á Á ; g r g of an Euclidean Jordan algebra A and considering the spaces A ii ¼ fx 2 A : L Å ðg i Þx ¼ xg and the spaces A ij ¼ fx 2 A : L Å ðg i Þx ¼ 1 2 x^L Å ðg j Þx ¼ 1 2 xg then we obtain the decomposition of A as an orthogonal direct sum of the vector spaces A ii s and A ij s in the following way: In the case when the Euclidean symmetric Jordan algebra is A ¼ Sym ðn; RÞ and we consider the Jordan frame of A, S ¼ fE 11 ; E 22 ; Á Á Á ; E nn g we obtain the following spaces A ii ¼ fA 2 M n ðRÞ : 9a 2 R : A ¼ aE ii g and the spaces A ij ¼ fA 2 M n ðRÞ : 9x ij 2 R : A ¼ x ij ðE ij þ E ji Þg; where the matrices E ii s are the matrices with 1 in the entry ii and with the others entries zero and the matrix E ij is the matrix with 1 in the entry ij and zero on the others entries. Therefore the

Pierce decomposition of A relatively to the Jordan frame of
So, we conclude that A nþ1 ðc; 1Þ ¼ fac; a 2 Rg: Now, we will calculate A nþ1 ðc; 0Þ:

À Á
: Therefore, we obtain, Now, we will analyse the Pierce decomposition of A nþ1 relatively to the Jordan frame As in the previous example, we have ðA nþ1 Þ 11 ¼ A nþ1 ðc 1 ; 1Þ ¼ fac 1 : a 2 Rg; and ðA nþ1 Þ 22 ¼ A nþ1 ðc 2 ; 1Þ ¼ fac 2 : a 2 Rg: Now, we will calculate the vector space So, we have the following calculations 5 : x 2 R nÀ1 g: Now, we will calculate A nþ1 c 2 ; 1 2 À Á : Therefore for any element x ¼

5:
A brief introduction to strongly regular graphs An undirect graph X is a pair of sets (V(X), ; v n g and E(X), the set of edges of X, a subset of V(X) Â V(X). For simplicity, we will denote an edge between the vertices a and b by ab. The order of the graph X is the number of vertices of X ; jV ðX Þj and we call the dimension of X ; jEðX Þj to the number of edges of X. One calls a graph a simple graph if it has no multiple edges (more than one edge between two vertices) and if it has no loops.
Sometimes we make a sketch to represent a graph X like the one presented on the Figure 1. An edge is incident on a vertice v of a graph X if v is one of its extreme points. Two vertices of a graph X are adjacent if they are connected by an edge. The adjacency matrix of a simple graph X of order n is a square matrix of order n, A such that A ¼ ½a ij where a ij ¼ 1 if v i v j 2 EðX Þ and 0 otherwise. The adjacency matrix of a simple graph is a symmetric matrix and we must observe that the diagonal entries of this matrix are null. The number of edges incident to a vertice v of a simple graph is called the degree of v. And, we call a simple graph a regular graph if each of its vertices have the same degree and we say that a graph G is a k-regular graph if each of its vertices have degree k.
The complement graph of a graph X denoted by X is a graph with the same set of vertices of X and such that two distinct vertices are adjacent vertices of X if and only if they are non adjacent vertices of X.
Along this paper we consider only non-empty, simple and non complete graphs. Strongly regular graphs were firstly introduced by R. C. Bose in the paper [33].
A graph X is called a (n, k; k, l)-strongly regular graph if is k-regular and any pair of adjacent vertices have k common neighbors and any pair of non-adjacent vertices have l common adjacent vertices.
The adjacency matrix A of a (n, k; k, l)-strongly regular X satisfies the equation A 2 ¼ kI n þ kA þ lðJ n À A À I n Þ, where J n is the all ones real matrix of order n.
The eigenvalues h, s, k and the multiplicities m h and m s of h and s respectively of a (n, k; k, l)-strongly regular graph X, see, for instance [34,35], are defined by the equalities (5): Therefore necessary conditions for the parameters of a (n, k; k, l)-strongly regular graph are that jsjnþsÀk nðhÀsÞ and hnþkÀh nðhÀsÞ must be natural numbers, they are known as the integrability conditions of a strongly regular graph, see [34]. We note that a graph X is a (n, k; k, l)-strongly regular graph if and only if its complement graph X is a (n, n À k À 1; n À 2 À 2k + l, n À 2k + k). Now we will present some admissibility conditions on the parameters of a (n, k; k, l)-strongly regular graph. The parameters of a (n, k; k, l)-strongly regular graph X verifies the admissibility condition (6).
The inequalities (7) are known as the Krein conditions of X, see [36].
Given a graph X, we call a path in X between two vertices v 1 and v k+1 to a non null sequence of distinct vertices, exceptionally the first vertice and the last vertice can be equal, and distinct edges, W = v 1 e 1 v 2 e 2 v 3 Á Á Áv k e k v k+1 whose terms are vertices and edges alternated and such that for 1 i k the vertices v i and v iþ1 define the edge e i . The path is a closed path or a cycle if and only if the only repeated vertices are the initial vertice and the final vertice. A graph Y 0 is a subgraph of a graph Y and we We must observe that for any non empty subset V 0 of V(Y) we can construct a subgraph of Y whose set of vertices is V 0 and whose set of edges is formed by the edges of E(Y) whose extreme points are vertices in V 0 which we call the induced subgraph of Y and which we denote by Y ðV 0 Þ: Two vertices v 1 and v 2 of a graph X are connected if there is a path between v 1 and v 2 in X. This relation between vertices is a relation of equivalence in the set of vertices of the graph X, V(X), whereby there exists a partition of V(X) in non empty subsets V 1 , V 2 ,Á Á Á, V I of V(X) such that two vertices are connected if and only if they belong to the same set V i for a given i 2 f1; 2; Á Á Á ; lg: The subgraphs X (V 1 ), X(V 2 ), Á Á Á, X(V l ) are called the connected components of X. If X has only one component then we say that the graph X is connected otherwise the graph X is a disconnected graph. A (n, k; k, l)-strongly regular graph X is primitive if and only X and X are connected graphs. Otherwise is called an imprimitive strongly regular graph. To characterize the connected graphs we will present the definition of reducible and of irreducible matrix. Let n be a natural number greater or equal 2 and A a matrix in M n ðRÞ: We say that the matrix A is a reducible matrix if there exists a permutation matrix P such that where k is such that 1 k nÀ1, if doesn't exist such matrix P we say that the matrix A is irreducible. If A is a reducible symmetric of order n then D nÀk Âk ¼ O nÀk Âk : From the Theorem of Frobenius we know that if A is a real square irreducible matrix with non negative entries then A has an eigenvector u such that Au = ru with all entries positive and such that jkj r for any eigenvalue of A and r is a simple positive eigenvalue of A. Now since a graph is connected if and only if its matrix of adjacency is irreducible then, if a graph X is a connected strongly regular graph then the greater eigenvalue of its adjacency matrix A is a simple eigenvalue of A with an eigenvector with all components positive. Hence the regularity of a connected strongly regular graph X is a simple eigenvalue of the adjacency matrix of X: Finally, since from now on, we only consider primitive strongly regular graphs, we note that a (n, k; k, l)-strongly regular graph is imprimitive if and only if l = 0 or l = k. From now on, we only consider (n, k; k, l)strongly regular graphs with k À 1 ! l > 0. The multiplicities of the eigenvalues h and s of a primitive strongly regular graph X of order n satisfy the conditions (10) known has the absolute bounds and they also satisfy the equalities 1 þ m h þ m s ¼ n and k þ m h h þ m s s ¼ 0:

Some relations on the parameters of a strongly regular graph
Let m be a natural number. We denote the set of real matrices of order m by M m ðRÞ and the set of symmetric of M m ðRÞ by Sym ðm; RÞ. For any two matrices H ¼ ½h ij and L ¼ ½l ij of M m ðRÞ: we define the Hadamard product of H and L as being the matrix H L ¼ ½h ij l ij and the Kronecker product of matrices H and L as being the matrix H L ¼ ½h ij L: For any matrix P of M m ðRÞ and for any nonnegative integer number j we define the Schur (Hadamard) power of order j of P, as being the matrix P j in the following way: P 0 ¼ J n ; P 1 ¼ P for any natural number j ! 2 we define P jþ1 ¼ P P j : In this section we will establish some inequalities over the parameters and over the spectra of a primitive strongly regular graph.
Let's consider a primitive (n, k; k, l)-strongly regular graph G such that n 2 > k > l > 0 and with l < k, and let A be its adjacency matrix with the distinct eigenvalues s, h and k: Now, we consider the Euclidean Jordan algebra A ¼ Sym ðn; RÞ with the Jordan product x Å y ¼ xyþyx 2 and with the inner product xjy ¼ trace ðx Å yÞ; where xy and yx are the usual products of x by y and the usual product of y by x. Now we consider the Euclidean Jordan subalgebra A of Sym ðn; RÞ spanned by I n and the natural powers of A. We have that rank ðAÞ ¼ 3 since has three distinct eigenvalues and is a three dimensional real Euclidean Jordan algebra. Let B ¼ fE 1 ; E 2 ; E 3 g be the unique Jordan frame of A associated to A, where Þ ; E 2 ¼ jsjn þ s À k nðh À sÞ I n þ n þ s À k nðh À sÞ A þ s À k nðh À sÞ ðJ n À A À I n Þ; Let's consider the real positive number x such that x 1, and let's consider the binomial Hadamard series l Àz l ðA 2 Àh 2 InÞ 2 k 4 l : The second spectral decomposition of S z relatively to the Jordan frame B is q iz E i : Now, we show that the eigenvalues q iz of S z are positive.
Since ðÀ1Þ l Àz l ¼ ðÀ1Þ l ðÀzÞðÀz À 1ÞðÀz À 2Þ Á Á Á ðÀz À l þ 1Þ l! then l Àz l ðA 2 Àh 2 InÞ 2 k 4 l : Since A 2 ¼ kI n þ kA þ lðJ n À A À I n Þ then we conclude that Let's consider the second spectral decomposition S nz ¼ q n1z E 1 þ q n2z E 2 þ q n3z E 3 : Since k > l then we have that jsj < h and therefore jsj 2 < h 2 and since h 2 k 2 then the eigenvalues of A 2 Àh 2 In k 4 are positive. Since for any two matrices C and D of M n ðRÞ we have k min ðCÞk min ðDÞ k min ðC DÞ and since B is a Jordan frame of A that is a basis of A and A is closed for the Schur product of matrices we deduce that the eigenvalues of ðA 2 Àh 2 InÞ 2 k 4 l are positive. So we conclude that the eigenvalues q niz for i = 1, Á Á Á, 3 of S nz are all positive. Since q 1z ¼ lim n!þ1 q n1z ; q 2z ¼ lim n!þ1 q n2z ; q 3z ¼ lim n!þ1 q n3z then we have q 1z ! 0; q 2z ! 0 and q 3z ! 0: We must say k 4 Þ z ðÀs À 1Þ: Let's consider the element S 3z ¼ E 3 S z of A: Since the eigenvalues of E 3 and of S z are positive and since k min ðE 3 Þk min ðS z Þ k min ðE 3 S x Þ then the eigenvalues of E 3 S z are positive. Now since k < n 2 and k > l and by an asymptotical analysis of the spectrum of E 3 S z we will deduce the inequalities (16) and (21) of the Theorems 3 and 4 respectively are verified. Now, we consider the second spectral decomposition E 3 S z ¼ q 1 3z E 1 þ q 2 3z E 2 þ q 3 3z E 3 : Then, we have Now since when k < n 2 we have nÀkþh hnþkÀh > 1 2hþ1 and after some algebraic manipulation of the inequality (19) we obtain the inequality (20).
And, finally we conclude that k 4 Àl 2 k 4 ÀðkÀh 2 Þ 2 2hþ1 > k 4 Àl 2 k 4 Àk 2 k : Theorem 4. Let n, k, l and k be natural numbers and let X be a (n, k; k, l)-primitive strongly regular graph with n À 1 > k > l; n 2 > k; l < k and with the distinct eigenvalues h, s and k: Then Proof. Now since q 3 3z ! 0 and recurring to the equality (15) From (22) we deduce the inequality (23).
Calculating the limits of the expressions of both hand sides of (23) when x approaches zero we obtain the equality (24).

Preliminares on quaternions and octonions
This section is a brief introduction on quaternions and octonions. Good readable texts on this algebraic structures can be found on the works [37,38]. Now, we consider the real linear space A of quaternions spanned by the basis B ¼ f1; i; j; kg, where the elements of B verify the following rules of multiplication i 2 ¼ j 2 ¼ k 2 ¼ À1 and 1) ij ¼ Àji ¼ k; 2) jk ¼ Àkj ¼ i; 3) ki ¼ Àik ¼ j: So, we can write A ¼ fa 0 1 þ a 1 i þ a 2 j þ a 3 k; a 0 ; a 1 ; a 2 ; a 3 2 Rg: Given a quaternion x ¼ x 0 1 þ x 1 i þ x 2 j þ x 3 k we call to x 0 the real part of x we denote it by Re (x) and we call to x 1 i þ x 2 j þ x 3 k the imaginary part of x and we write Im ðxÞ ¼ x 1 i þ x 2 j þ x 3 k: One says that a quaternion x is a pure quaternion if Re (x)=0 and if x = Re (x) one says that the quaternion x is a real number.
For discovering the equalities for multiplication describe above we must use the diagram of Fano, see Figure 2.
When we multiply to elements of the set fi; j; kg we use the rule: when we multiply two elements in clockwise sense we get the next element, so for instance we have jk = i, but if we multiply them in the counterclockwise sense we obtain the next but with minus sign kj = Ài.