© L. Vieira, Published by EDP Sciences, 2019

Introduction

The Euclidean Jordan algebras have many applications in several areas of mathematics. Some authors applied the theory of Euclidean Jordan algebras to interior-point methods [1–10], others applied this theory to combinatorics [11–14], and to statistics [15–17]. More actually, some authors extended the properties of the real symmetric matrices to the elements of simple real Euclidean Jordan algebras, see [18–24].

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 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 over a field with characteristic ≠ 2 is a vector space over the field with a operation of multiplication ⋄ such that for any x and y in , x ⋄ y = y ⋄ x and x ⋄ (x ^2⋄ ⋄ y) = x ^2⋄ ⋄ (x ⋄ y), where x ^2⋄ = x ⋄ x. We will suppose throughout this paper that if is a Jordan algebra then has a unit element that we will denote it by e.

When the field 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 ⋄.

The real vector space of real symmetric matrices, , of order n, with the operation is a real Jordan algebra.

We must note, that we define the powers of an element x in in the usual way and for any natural number k. Hence, we have and therefore by induction over we conclude that for any natural number k, where x ^k represents the usual power of order k of a squared symmetric matrix. is a Jordan algebra since for x and y in we have and

Let’s consider another example of Euclidean Jordan algebra. Let consider the real vector space equipped with the product ⋄ such that

where and Now we will show that and that We have

Herein, we must say that the element is the unit of the Euclidean Jordan algebra Indeed, we have

Now since the operation ⋄ is commutative, we showed that

Hence, e is a unit of the Jordan algebra Now, we have

and

Therefore, since for any z and w in we have z ⋄ w = w ⋄ z and . Then we conclude that is a Jordan algebra.

Now we define a Jordan subalgebra of the Euclidean Jordan algebra . Let’s consider the set of symmetric matrices of such that

1. B ₁ = I _n ,

2. , for and for i = 1,⋯, l,

3. B ₁ + B ₂+⋯+B _l = J _n ,

4. .

Then the real vector space spanned by the set with the Jordan product ⋄ is a Jordan algebra. Firstly, we must say that is closed for the Jordan product. Indeed, for C and D in we have and Therefore

and

Since then

So is a symmetric matrix of , but from property iv) we conclude that therefore in Now, we must say, that for any matrices X and we have X ⋄ Y = Y ⋄ X and since is a subalgebra of .

Let be a n-dimensional Jordan algebra. Then is power associative, this is, is an algebra such that for any x in the algebra spanned by x and e is associative. For x in we define rank (x) as being the least natural number k such that is a linearly dependent set and we write rank (x) = k. Now since then The rank of A is defined as being the natural number An element x in is regular if Now, we must observe that the set of regular elements of is a dense set in . Let’s consider a regular element x of and r = rank (x).

Then, there exist real numbers and such that

(1)where 0 is the zero vector of . Taking into account (1) we conclude that the polynomial p(x, −) define by the equality (2).

(2)is the minimal polynomial of x. When x is a non regular element of the minimal polynomial of x has a degree less than r. The polynomial p(x, −) is called the characteristic polynomial of x. Now, we must say that the coefficients are homogeneous polynomials of degree i on the coordinates of x on a fixed basis of . Since the set of regular elements of is a dense set in then we extend the definition of characteristic polynomial to non regular elements of by continuity. The roots of the characteristic polynomial p(x, −) of x are called the eigenvalues of x. The coefficient of the characteristic polynomial p(x, −) is called the trace of x and we denote it by Trace (x) and we call the coefficient the determinant of x and we denote it by Det (x).

Let be a real finite dimensional associative algebra with the bilinear map . We introduce on a structure of Jordan algebra by considering a new product • defined by for all x and y in . The product · is called the Jordan product of x by y. Let be a real Jordan algebra and x be a regular element of Then we have We define the linear operator such that We define the real vector space by The restriction of the linear operator to we call We must note now that and

A Jordan algebra is simple if and only if does not contain any nontrivial ideal. An Euclidean Jordan algebra is a Jordan algebra with an inner product ·|· such that , for all x, y and z in 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 equipped with the Jordan product 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 for x and y in is an Euclidean Jordan algebra. Indeed, we have

Now, we will show that the is an Euclidean Jordan algebra relatively to the inner product We have the following calculations

and

Hence, we conclude that And therefore is an Euclidean Jordan algebra.

Now, we will analyse the rank of the Euclidean Jordan algebra . Let consider the element x of with n distinct eigenvalues λ ₁, λ ₂,…, λ _n−1 and λ _n , and an orthonormal basis of of eigenvectors of x such that for i=1,⋯, n. Considering the notation e = I _n , then we have:

Therefore, the set is a linearly independent set of if and only if the set

is a linearly independent set of But, since

then the set is a linearly independent set of and therefore the set

is a linearly independent set of The set is a linear dependent set of since the set

is a linearly dependent set of because the dimension of is n. Therefore, we conclude that rank (x) = n.

Let x be an element of with k distinct non null eigenvalues λ _i s, and let and be an orthonormal basis of eigenvectors of x of the eigenvector space of x associated λ _i , this is Now, we consider the elements and we have Therefore, we have

and, since

then the set is a linearly independent set of And, the set is a linearly dependent set of since and therefore If x has k distinct eigenvalues where k−1 eigenvalues are non null and one is null then one proves one a similar way that .

Therefore we conclude that and the regular elements of are the elements x of with n distinct eigenvalues.

The characteristic polynomial of a regular element of is a monic polynomial of minimal degree Now, let x be an element of with n distinct eigenvalues λ ₁, λ ₂,⋯, λ _n−1 and λ _n then by the Theorem of Cayley-Hamilton we conclude that the polynomial p such that

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, −) is such that This is So, we have Therefore, we have and .

Now, we will show that To came to this conclusion, we will firstly show that for and that for So, let suppose then, we have

Therefore, the set is a linearly independent set of Now, we have and since

we conclude that If we have Then the set is a linearly dependent set of then And, therefore And the regular elements of are the elements of such that

Since, when we have

Then, supposing and considering the notation we conclude that the caractheristic polynomial of x is Therefore, the eigenvalues of x are and and

Let be a real Euclidean Jordan algebra with unit element e. An element f in is an idempotent of if . Two idempotents f and g of are orthogonal if A set of nonzero idempotents is a complete system of orthogonal idempotents of if , for i = 1, …, k, , for , and . An element g of is a primitive idempotent if it is a non null idempotent of and if cannot be written as a sum of two orthogonal nonzero idempotents of . We say that is a Jordan frame of if is a complete system of orthogonal idempotents such that each idempotent is primitive.

Let consider the matrices E _jj of the Euclidean Jordan algebra with j = 1,⋯, n where E _jj is the square matrix of order n such that (E _jj)_jj = 1 and (E _jj ) _ik  = 0 if i ≠ j or k ≠ j.

Let k be a natural number such that 1 < k < n. Then is a complete system of orthogonal idempotents of and is a Jordan frame of .

Let consider the Euclidean Jordan algebra and x non zero element of . Then the set is a Jordan frame of the Euclidean Jordan algebra Indeed, we have: and

and, we have

Therefore we conclude that is a Jordan frame of since

Theorem 1. ([28], p. 43). Let be a real Euclidean Jordan algebra. Then for x in there exist unique real numbers all distinct, and a unique complete system of orthogonal idempotents such that

(3)

The numbers ’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 be a real Euclidean Jordan algebra with Then for each x in there exists a Jordan frame and real numbers and such that

(4)

Remark 1. The decomposition (4) is called the second spectral decomposition of x. And we have and

Example 1. For , the second spectral decomposition of relatively to the Jordan frame of is

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:

1. The spin Euclidean Jordan algebra .

2. The Euclidean Jordan algebra with the Jordan product of matrices and with an inner product of two symmetric matrices as being the trace of their Jordan product.

3. The Euclidean Jordan algebra of hermitian matrices of complexes of order n equipped with the Jordan product of two hermitian matrices of complexes and with the scalar product of two Hermitian matrices of complexes as being the real part of the trace of their Jordan product.

4. The Euclidean Jordan algebra , of hermitian matrices of quaternions of order n equipped with the Jordan product of hermitian matrices of quaternions and with the scalar product of two hermitian matrices of quaternions as being the real part of the trace of their the Jordan product.

5. The Euclidean Jordan algebra of hermitian matrices of octonions of order n equipped with the Jordan product of two hermitian matrices of octonions and with the inner product of two hermitian matrices of octonions as being the real part of 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 the eigenvalues of the linear operator are and 1 and this fact permits us to say that, considering the eigenspaces and of L(g) associated to these eigenvalues we can decompose as an orthogonal direct sum Now, we will describe the Pierce decomposition of the Euclidean Jordan algebra relatively to an idempotent of the form .

Let and Now, we will show that We have

Therefore, we have

Now, let calculate So, let consider the following equivalences.

Hence, we have Now, let’s calculate . So let consider the following calculations:

Hence we have .

For, other hand if we consider a Jordan frame of an Euclidean Jordan algebra and considering the spaces and the spaces then we obtain the decomposition of as an orthogonal direct sum of the vector spaces s and s in the following way:

In the case when the Euclidean symmetric Jordan algebra is and we consider the Jordan frame of , we obtain the following spaces and the spaces 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 relatively to the Jordan frame of is Therefore, we can write, any matrix of the in the form

Let consider the spin Euclidean Jordan algebra and let consider the idempotent with We, will obtain the spaces , and We have, the following equivalences

So, we conclude that Now, we will calculate

Therefore, we obtain, We, can rewrite Now, we will obtain So, we have

Therefore, we obtain,

Now, we will analyse the Pierce decomposition of relatively to the Jordan frame

As in the previous example, we have and Now, we will calculate the vector space

So, we have the following calculations

Therefore Now, we will calculate

Hence, we have also that Therefore we have

Therefore for any element of we conclude that

A brief introduction to strongly regular graphs

An undirect graph X is a pair of sets (V(X), E(X)) with and E(X), the set of edges of X, a subset of V(X) × 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 and we call the dimension of 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.

Figure 1 A simple graph X.

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 where if 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 is a graph with the same set of vertices of X and such that two distinct vertices are adjacent vertices of 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; λ, μ)-strongly regular graph if is k-regular and any pair of adjacent vertices have λ common neighbors and any pair of non-adjacent vertices have μ common adjacent vertices.

The adjacency matrix A of a (n, k; λ, μ)-strongly regular X satisfies the equation , where J _n is the all ones real matrix of order n.

The eigenvalues θ, τ, k and the multiplicities m _θ and m _τ of θ and τ respectively of a (n, k; λ, μ)-strongly regular graph X, see, for instance [34, 35], are defined by the equalities (5):

(5)

Therefore necessary conditions for the parameters of a (n, k; λ, μ)-strongly regular graph are that and 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; λ, μ)-strongly regular graph if and only if its complement graph is a (n, n − k − 1; n − 2 − 2k + μ, n − 2k + λ). Now we will present some admissibility conditions on the parameters of a (n, k; λ, μ)-strongly regular graph. The parameters of a (n, k; λ, μ)-strongly regular graph X verifies the admissibility condition (6).

(6)

The inequalities (7) are known as the Krein conditions of X, see [36].

(7)

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⋯v _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 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′ is a subgraph of a graph Y and we write if and If and we say that Y′ is a proper subgraph of Y. We must observe that for any non empty subset V′ of V(Y) we can construct a subgraph of Y whose set of vertices is and whose set of edges is formed by the edges of E(Y) whose extreme points are vertices in V′ which we call the induced subgraph of Y and which we denote by 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,⋯, 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 The subgraphs X(V ₁), X(V ₂), ⋯, 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; λ, μ)-strongly regular graph X is primitive if and only X and 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 matrix in We say that the matrix is a reducible matrix if there exists a permutation matrix such that

(8)where k is such that 1 ≤ k ≤ n−1, 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 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 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; λ, μ)-strongly regular graph is imprimitive if and only if μ = 0 or μ = k. From now on, we only consider (n, k; λ, μ)- strongly regular graphs with k − 1 ≥ μ > 0. The multiplicities of the eigenvalues θ and τ of a primitive strongly regular graph X of order n satisfy the conditions (10) known has the absolute bounds

(9)

(10)and they also satisfy the equalities and

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 and the set of symmetric of by . For any two matrices and of we define the Hadamard product of H and L as being the matrix and the Kronecker product of matrices H and L as being the matrix For any matrix P of and for any nonnegative integer number j we define the Schur (Hadamard) power of order j of P, as being the matrix in the following way: for any natural number j ≥ 2 we define

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; λ, μ)-strongly regular graph G such that and with μ < λ, and let A be its adjacency matrix with the distinct eigenvalues τ, θ and k Now, we consider the Euclidean Jordan algebra with the Jordan product and with the inner product 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 of spanned by I _n and the natural powers of A. We have that since has three distinct eigenvalues and is a three dimensional real Euclidean Jordan algebra. Let be the unique Jordan frame of associated to A, where

Let’s consider the real positive number x such that x ≤ 1, and let’s consider the binomial Hadamard series The second spectral decomposition of S _z relatively to the Jordan frame is Now, we show that the eigenvalues q _iz of S _z are positive.

Since then

Now, we have Since then we conclude that

Let’s consider the second spectral decomposition Since λ > μ then we have that and therefore and since then the eigenvalues of are positive. Since for any two matrices C and D of we have and since B is a Jordan frame of that is a basis of and is closed for the Schur product of matrices we deduce that the eigenvalues of are positive. So we conclude that the eigenvalues q _niz for i = 1, ⋯, 3 of S _nz are all positive.

Since then we have and We must say that and hence we have: and

Let’s consider the element of Since the eigenvalues of E ₃ and of S _z are positive and since then the eigenvalues of are positive. Now since and λ > μ and by an asymptotical analysis of the spectrum of we will deduce the inequalities (16) and (21) of the Theorems 3 and 4 respectively are verified.

Now, we consider the second spectral decomposition Then, we have

(11)

(12)

(13)

Now, since then and consequently the parameter takes the form:

(14)and takes the form:

(15)

Using the fact that then we conclude that

Theorem 3. Let n, k, μ and λ be natural numbers with n − 1 > k > μ and X be a (n, k; λ, μ)-primitive strongly regular graph with distinct eigenvalues k, θ and τ If and then

(16) Proof. Since and recurring to the equality (14) we deduce the equality (17).

(17)

Making, some algebraic manipulation of equality (17) we obtain the inequality (18).

(18)

Calculating the limits of the expressions of both hand sides of (18) when x approaches zero we obtain the equality (19).

(19)

Now since when we have and after some algebraic manipulation of the inequality (19) we obtain the inequality (20).

(20)

And, finally we conclude that

Theorem 4. Let n, k, μ and λ be natural numbers and let X be a (n, k; λ, μ)-primitive strongly regular graph with and with the distinct eigenvalues θ, τ and k Then

(21) Proof. Now since and recurring to the equality (15) we obtain that

(22)

From (22) we deduce the inequality (23).

(23)

Calculating the limits of the expressions of both hand sides of (23) when x approaches zero we obtain the equality (24).

(24)

Now, since when we have and after some algebraic manipulation of the inequality (24) we obtain the inequality (25).

(25)And, finally we conclude that

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 of quaternions spanned by the basis , where the elements of B verify the following rules of multiplication and

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.

Figure 2 Diagram of Fano for quaternions.

When we multiply to elements of the set 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 = −i.

And, therefore we obtain Table 1 of multiplication of two elements of A.

If we consider and we obtain the following expression for the product x y,

Using the notation and we conclude that

For a quaternion we define we define by the the equality Now, we will show that Hence, we have the following calculations:

In a similar way, we deduce that The inverse of a nonzero quaternion x is define as an element x ⁻¹ such that x x ⁻¹ = x ⁻¹ x = 1. But since where x ^2* = x x. then,

Now, we will introduce the real linear space of octonions. where the elements of the basis of satisfy the rules of multiplication presented in Table 2 below and deduced using the diagram presented in Figure 3.

Figure 3 Diagram of Fano for octonions.

Now, we fulfill the table recurring to the diagram of Fano, for instance we have f ₂ f ₄ = f ₆ but we have f ₄ f ₂ = −f ₆ since the sense from f ₄ to f ₂ is contrary to the sense of line that contains f ₄, f ₂ and f ₆.

The conjugate of the octonion is and the real part of the octonion x is and the imaginary part of the octonion is We define

Using the table of multiplication 2 we obtain for and for we obtain

We must note that: Proceeding like we have done for the quaternions we deduce that We define the norm of an octonion as being We, must say, again that Let x be an octonion such that then the inverse of x is since x x ⁻¹ = x ⁻¹ x = 1.

Acknowledgments

Luís Vieira was partially supported by CMUP (UID/MAT/00144/2019), which is funded by FCT with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT 2020.

The author are grateful to the anonymous referee for a careful checking of the details and for helpful comments that improved this paper.

References

All Tables

All Figures

	Figure 1 A simple graph X.
In the text

	Figure 2 Diagram of Fano for quaternions.
In the text

	Figure 3 Diagram of Fano for octonions.
In the text