Binomial Hadamard Series and Inequalities over the Spectra of a Strongly Regular Graph ()
1. Introduction
Good surveys on Euclidean Jordan algebras can be found in books such as A Taste of Jordan Algebras of Kevin McCrimmon [1] , Analysis on Symmetric Cones of Faraut and Korányi [2] and in the Koecher’s Minnesota Notes on Jordan Algebras and Their Applications [3] . Euclidean Jordan algebras become a good tool for the analysis of primal dual interior point methods [4] [5] [6] and [7] . But they have also a lot of applications on other branches of mathematics namely on the formalism of quantum mechanics [8] , on combinatorics [9] - [15] , and on statistics [16] . Recently, many authors had extended some properties of matrix theory to the Euclidean Jordan Algebras, see for example [17] [18] and [19] .
In this paper we analyse the spectra of Hadamard series associated to the adjacency matrix of a strongly regular graph to deduce asymptotic inequalities on the spectra and on the parameters of a strongly regular graph in the environment of Euclidean Jordan algebras.
The plan of this paper is as follows. In Section 2 we expose the principal definitions and results on Euclidean Jordan algebras. Next, in Section 3 we present the more relevant definitions and results on strongly regular graphs necessary for a clear exposition of this paper. Finally, in Section 4 we associate a three dimensional real Euclidean Jordan algebra A to a strongly regular graph and next we establish asymptotic inequalities on the spectra and on the parameters of a strongly regular graph, see (10) and (20) of Theorem 4 and 5 respectively.
2. A Short Introduction to Euclidean Jordan Algebras
Herein, we make an introduction to Euclidean Jordan algebras and present the definitions and the more relevant results needed for this paper without presenting the proofs of the results presented in this paper, since they are very well deduced in the monograph Analysis on symmetric cones of Faraut and Korányi, see [2] .
Let
be a vector space of finite dimension over a field
with the bilinear mapping
from
to
. Then,
is called a power associative algebra if for any
the subalgebra of
spanned by the powers of x is associative. If for all u and v in
we have (
)
and (
)
, with
, then
is called a Jordan algebra.
If
is a Jordan algebra with unit element, then
is power associative (cf. [2] ).
Example 1. Let n be a natural number and A and B be two real matrices of order n. Then it is well known that in general
where AB and BA represents the usual products of the matrices A by B and the usual product of the matrix B by A. Nevertheless, the multiplication
define such that
for all real matrices A and B of order n verifies
. Now, we will show that, for any two real square matrices
and
of order n, we have
. Indeed, we have that
And, therefore we have
. Hence, the vector space
over the field of the real numbers with the operation
is a Jordan algebra. But, we must also say that with this operation,
, the square of a real matrix of order n is such that
where AA represents the usual product of matrices. One proves by induction that the power of order n relatively to the product
of a matrix
is equal to the usual power of order n of A. The subalgebra
of
formed by the real symmetric matrices of order n of
with the operation
is also a Jordan algebra.
Remark 1. If
is a vector space over a field
with characteristic distinct from 2 with the operation
from
onto
such that
with this operation is an associative algebra, then
becomes a Jordan algebra when
equipped with the operation
such that1
.
From now on, we only deal with real Jordan algebras with finite dimension and with unit and we will denote it by
, and when we say let
be a Jordan algebra we are meaning that we are saying that
is a real Jordan algebra with the element
and is of finite dimension.
Let
be a Jordan algebra with the operation
of multiplication of two elements of
with unit element e. If there is an inner product
on
such that
, for any u, v, w in
then
is called an Euclidean Jordan algebra.
Sometimes, we will say let
be an Euclidean Jordan algebra and we will use the notation
to represent the product of the elements x and y and
to represent the inner product of x and y. Or, we will say let
be an Euclidean Jordan algebra.
Remark 2. Let
be an algebra over a field
equipped the operation of multiplication
from
to
and with a inner product
. Sometimes one defines for any
the linear operator
to express that
is an Euclidean Jordan algebra if and only if
and
and
.
Some examples of Euclidean Jordan algebras over the real numbers are:
1) the n dimensional Euclidean Jordan algebra
endowed with the product
and
defined in the following way: for
and for
and
. The unit element of
is
.
2) the Euclidean Jordan algebra
, with
and with the operations
and
defined in the following way: for
and
,
and
. The unit element of
is
.
3) the Euclidean Jordan algebra of real symmetric matrices of order n,
endowed with the Jordan product
and the
inner product defined by
, where tr denotes the usual trace of matrices. The unit element of
is the identity matrix of order n,
.
4) The Euclidean Jordan algebra of self adjoint complex matrices of order n,
with
and
for x and
.
5) the Euclidean Jordan algebra of self adjoint quaternionic matrices of order n,
with
and
for x and
.
6) the Euclidean Jordan algebra of self adjoint octonionic matrices of order 3,
with
and
for x and
.
From now on, we will suppose that when we say let
be an Euclidean Jordan algebra we suppose that
is a real finite dimensional Euclidean Jordan algebra with unit element, that we will denote by e and for a more explicit writing sometimes we will say let
be an Euclidean Jordan algebra for meaning that
is an Euclidean Jordan algebra with the operation
of two elements of
and equipped with the inner product
and that
has the unit element e.
Let consider the Euclidean Jordan algebra
. An element
is an idempotent if
. Two idempotents c and d in
are orthogonal if
. Herein, we must say that if two elements of the Euclidean Jordan algebra
are orthogonal relatively to the product
of the Euclidean Jordan algebra
then they are also orthogonal relatively to the inner product of this Euclidean Jordan algebra. Indeed, if
then we have
Let l be a natural number. The set
is a complete system of orthogonal idempotents if the following three conditions hold: 1)
, for
, 2)
, if
, and 3)
. An idempotent f is primitive if it is a non-zero idempotent of
and cannot be written as a sum of two orthogonal non-zero idempotents. We say that
is a Jordan frame if
is a complete system of orthogonal idempotents such that each idempotent is primitive.
Example 2. Let consider the Euclidean Jordan algebra
with
the Jordan product
and the inner product
. We will show that
is a complete system of orthogonal idempotents of
. First, we must note that
.
So the Jordan square of an element of
is the usual square of a symmetric matrix. Now since
,
,
and
, then
is a complete system of orthogonal idempotents of
, but S is not a Jordan frame of
since
with
and
,
and
and
and
.
Let consider the Euclidean Jordan algebra
. The rank of an element x in
is the least natural number l, such that the set
is linearly dependent (where
and
), and we write
. This concept is expanded by defining the rank of the algebra
as being the natural number
.
The elements of
with rank equal to the rank of
are the regular elements of
. This set of the regular elements of
is an open and dense set in
. If u is a regular element of
, with
, then the set
is linearly dependent and the set
is linearly independent. Thus, we may conclude that there exist unique real numbers
, such that
,
where 0 is the null vector of
. Making the necessary adjustments we obtain the polynomial in
,
such that
(1)
The polynomial
is called the characteristic polynomial of u, where each coefficient
is a homogeneous polynomial of degree i in the coordinates of u in a fixed basis of
. Although the characteristic polynomial
is defined for a regular element of
, we can extend the definition of characteristic polynomial to all elements of
by continuity since each polynomial
is homogeneous and the set of regular elements of
is dense in
. Herein, we must say that the characteristic polynomial of a regular element of
coincides with his minimal polynomial and for a non regular element the minimal polynomial of this element divides its characteristic polynomial.
Herein, we will justify the reason to call the polynomial
the characteristic polynomial of x when x is regular. Let x be a regular element of the Euclidean Jordan algebra
,
represent the restriction of the operator
to the space
and let
. Now since x is a regular element of
then we can say that
is a basis of
. Now, let express the images of the elements of the basis
by the linear application
on the basis
. Hence, we present the following calculations:
Therefore the matrix of the operator
on the basis
is
Now, we have
The roots of the characteristic polynomial of
and
, are called the eigenvalues of x. Furthermore, the coefficients
and
of the characteristic polynomial of x, are called the trace and the determinant of x, respectively. We most note that when x is regular element of
then
and that
.
Remark 3. For a clear understanding of the theory of Euclidean Jordan algebras a very good readable survey is the chapter 5, An Introduction to Formally Real Jordan Algebras and Their Applications in Optimization written by Farid Alizadeh in the book [20] .
Theorem 1. ( [2] , p. 43). Let
be a real Euclidean Jordan algebra. Then for u in
there exist unique real numbers
,
, all distinct, and a unique complete system of orthogonal idempotents
such that
(2)
The numbers
’s of (2) are the eigenvalues of u and the decomposition (2) is the first spectral decomposition of u.
Example 3. Let A be a matrix of the Euclidean Jordan algebra
If A is a matrix with l distinct eigenvalues
and
, then the complete system of orthogonal idempotents associated to A is the set
where each idempotents
for
is the projector on the eigenvector space of A associate to the eigenvalue
defined by the equalities (3)
(3)
for
. We must say, that they are unique. We first note that since
s are projectors then we have
,
then
and
for
and we have
is the first spectral decomposition associated to A.
But there is another way, for a clear exposition of this method of calculation see [20] , to construct the idempotents
’s on a more practically way. So will describe the method of construction of the idempotents. We have
. So we obtain the following system
(4)
Since the matrix of the coefficients of the system (4), considering
for
as unknowns and
and
as constants, is a non singular matrix since this is the transpose of the Vandermond matrix2, then using the Crammer rule we obtain that
for
.
Let now present a practical example. Let consider an element of the Euclidean
Jordan algebra
,
. We have that the characteristic polynomial of A is
. Let consider the notation
and
. Now a complete system of orthogonal idempotents associated to A is
with
Theorem 2. ( [2] , p. 44). Let
be a real Euclidean Jordan algebra with
. Then for each u in
there exists a Jordan frame
and real numbers
and
such that
(5)
The decomposition (5) is called the second spectral decomposition of u.
Remark 4. Now, let suppose that the n-finite dimensional real Euclidean Jordan algebra
verifies
. Since the set of regular elements of
is dense in
, then let x be one of his regular elements. So, there exists a unique complete system of orthogonal idempotents
such that
. But now we have that
is3 a linear independent set of
, therefore
is a basis of
.
Example 4. Let
be the Euclidean Jordan algebra such
equipped with the
operation such that
and equipped with
the inner product
and let B be a symmetric matrix of
and let
be an orthonormal basis of
of eigenvectors of B with
, for
. We suppose that we are using the column notation, this is we consider the notation
Let consider
for
. Then
is a Jordan frame of
. Indeed, let i be a natural number less or equal n, we have
Let i and j be two natural numbers such that
and such that
. Then we have
where
is the real null matrix of order n.
So, we have proved that two any distinct idempotents
s are orthogonal. Finally, since
is an orthonormal basis of
then we have
, therefore we have
. Hence, we have proved that
is a Jordan frame of the Euclidean Jordan algebra
. Finally, since
for
then we have
Then the second spectral decomposition of B is
.
3. Some Notions on Strongly Regular Graphs
A graph G non complete and non null is a strongly regular graph with parameters
if G is k-regular if for any two adjacent vertices of G have exactly
common neighbors and any two non-adjacent vertices have
common neighbors.
We will say, sometimes that G is a
strongly regular graph if G is a strongly regular graph with parameters
.
Let G be a
strongly regular graph. It is well known that the adjacency matrix of G,
, that is a binary matrix of order n such that
, if the vertex i is adjacent to j and 0 otherwise, satisfies the equation
,
where
is the all one matrix of order n. The eigenvalues of G, see for instance [21] , are k,
and
, where
and
are given by
and
, (see [21] ).
We must observe that G is a
strongly regular graph if and only if the complement graph of G,
is a
strongly regular graph. We must also note that the multiplicities of the eigenvalues
and
of G are:
and
respectively.
From now on, we will present some well known admissibility conditions on the parameters of a strongly regular graph.
We now present in Theorem 3 an admissibility relationship between the parameters of a strongly regular graph.
Theorem 3. Let G be a
strongly regular graph. Then
.
L.L. Scott in [22] established the Krein admissibility conditions (6) and (7).
(6)
(7)
Finally, we couldn’t help of presenting the admissibility conditions on the order of a strongly regular graph G and on the multiplicity of each eigenvalue distinct from the regularity of G, known as the absolute bounds introduced by Delsarte, Goethals and Seidel [23] , which we present on the inequalities (8) and (9).
(8)
(9)
A strongly regular graph G is called primitive if G and
are connected. A strongly regular graph that is not primitive is called an imprimitive strongly regular graph. But, we must say, that a
strongly regular graph G is imprimitive if and only if
or
, since we analyse only non complete strongly regular graphs then we can conclude that if G is a primitive strongly regular graph then
. In this paper we consider only primitive strongly regular graphs. In the section 4 we present some admissibility conditions on the spectra and on the parameters of a strongly regular graph but obtained on asymptotic algebraic way.
4. Binomial Hadamard Series and Inequalities over the Spectra of a Strongly Regular Raph
Let G be a
strongly regular graph with
and let A be its adjacency matrix with the distinct eigenvalues, namely k,
and
. Now let
be 3 dimensional real Euclidean subalgebra, with
, of the Euclidean Jordan algebra
spanned by
and the natural powers of A.
Now, we consider the unique complete system of orthogonal idempotents
of
associated to A, with
,
,
and
.
Let
be natural numbers such that
and
. So, since the idempotents
and
are orthogonal relatively to the Jordan product of matrices, then they are orthogonal relatively to the inner product
they are also orthogonal to the inner product
, for all
. Therefore we conclude that
is a basis of
.
Now, we will consider some notation for defining the Hadamard product and the Kronecker product of two matrices. We denote the space of real square matrices of order n by
and we define the Hadamard product and the Kronecker product of two matrices E and F of order n of
in the following way: if
and
, then we define
and
for all
. For any natural number l and for any matrix
we define
like as follows:
and for
we define
(see [24] ).
Let x be a real number such that
, herein we analyze different Hadamard series from the one used on the publication [9] but we use a algebraic method similar to the method that was used in the paper [9] . Let consider the binomial Hadamard series
.
Then
is the spectral decompositions of
respectively to the Jordan frame
of
. Now, we will show that the eigenvalues
s of
are positive. We have
and therefore
We denote the partial sum of order n of the Hadamard series
by
. Hence
and let consider the notation
.
Now we must note the eigenvalues of
are positive and that for any
two real matrices E and F of order n we have
, and we must also observe that since
is a Jordan frame of the Euclidean Jordan algebra
that is a basis of
and this Euclidean Jordan algebra is closed for the Hadamard product then we conclude that the eigenvalues of
are all positive.
Since
,
,
then we have
and
.
We have
and
, and
,
,
.
By an asymptotical analysis of the eigenvalues
and
we establish the inequalities (10) and (20) of Theorem 4 and of Theorem 5 respectively.
Theorem 4. Let G be a
-strongly regular graph with
and with the distinct eigenvalues
and
. If
then
(10)
Proof. Let write the binomial series
on the basis
of the Euclidean Jordan algebra
.
Since
then we have
(11)
Therefore, we conclude that
(12)
Since
then
(13)
Since
and
then rewriting (13) we obtain
.
And therefore
(14)
After some algebraic manipulation of (14) we obtain the inequality (15).
(15)
Now, applying limits as x tends to zero to both hand sides of (15) we obtain (16).
(16)
Recurring to the rule of L’Hopital to the second factor of the right hand side of (16) we obtain (17)
(17)
Recurring to the properties of logarithms on the right hand side of the inequality (17) we deduce (18).
(18)
Rewriting (18) we obtain the inequality (19)
(19)
Theorem 5. Let G be a
-strongly regular graph with
and with the distinct eigenvalues
and
. If
then
(20)
Proof. Since
,
and
then, we have (21).
(21)
After some algebraic manipulation of (21) we obtain (22).
(22)
After a rewriting of the inequality (22) we deduce (23).
(23)
Applying limits as x tends to zero to both hand sides of (23) we obtain inequality (24).
(24)
Applying the rule of l’Hopital to the second factor of the right hand side of (24) as x tends 0 we deduce (25).
(25)
Recurring to the properties of logarithms we deduce from (25) the inequality (26).
(26)
And, finally from (26) we conclude that
.
Now, considering the elements of
,
and
and analysing the eigenvalues
and
of the spectral decompositions
and
respectively, and making similars algebraic asymptotic analysis to that done on the proofs of Theorems 4 and 5 we deduce the inequalities (27) and (28) of Theorem 6.
Theorem 6. Let G be a
-strongly regular graph with
and with the distinct eigenvalues
and
. If
and
then
(27)
(28)
Acknowledgements
Lus Vieira in this work was partially supported by the Center of Research of Mathematics of University of Porto (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020.
NOTES
1We must note that
, since we have
2We must note that
3Note that any two distinct elements of
are orthogonal relatively to the inner product of the Euclidean Jordan algebra
.