Some Inequalities over the Eigenvalues of a Strongly Regular

Graph

Abstract: Let’s consider a primitive strongly regular graph G. In this paper, we establish some inequalities over

the spectrum of Gin the environment of a real ﬁnite dimensional Euclidean Jordan algebra Aassociated with G

recurring to a spectral analysis of some elements of Aand recurring to a spectral analysis of the Generalized Krein

parameters of G.

Key–Words:Euclidean Jordan Algebras, Theory of Graphs, Strongly regular graphs.

Received: November 21, 2022. Revised: May 15, 2023. Accepted: June 10, 2023. Published: July 14, 2023.

1 Introduction

For a precise and hard description of Euclidean

Jordan algebras, one must cite the monographs books,

“Analysis on Symmetric Cones” , [2] and “Strongly

Regular Graphs and Euclidean Jordan Algebras Rev-

elations within an Unusual Relationship”, [3].

Several mathematicians and scientists developed

several applications of the theory of real Euclidean

Jordan algebras in many areas of research, see for in-

stance [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]

and [14] but our main goal is, recurring to this theory,

to establish inequalities over the parameters of some

symmetric discrete structures like the strongly regular

graphs and the symmetric association schemes, see for

instance [15] and [16].

This paper is organized as follows. In the section

2 we present the main results about real ﬁnite dimen-

sional Euclidean Jordan algebras necessary for a good

understanding of the following sections, and present

in this section some examples. In the section 3 the

more relevant results about strongly regular graphs

are described. In section 4 some inequalities are es-

tablished over the eigenvalues of a primitive strongly

regular graph. Finally, in section 5 some conclusions

are presented.

2 Some Concepts about Euclidean

Jordan Algebras

In this section, we present some concepts about real

ﬁnite-dimensional Euclidean Jordan algebras and fol-

low the notation and the text presented in [1].

For a very readable text about the principal results

and clear examples of Jordan algebras and properties

of Euclidean Jordan algebras one must indicate the

work “Strongly Regular Graphs and Euclidean Jor-

dan Algebras Revelations within an Unusual Relation-

ship”, [3].

But we couldn’t avoid citing the chapter “Sym-

metric Cones, Potential Reduction Methods and word

by word Extensions”, see [17], for a very good intro-

ductory text about Euclidean Jordan algebras.

A real ﬁnite-dimensional Euclidean Jordan alge-

bra is a real ﬁnite-dimensional algebra, with an op-

eration of vector multiplication Fsuch that for any

of its elements, u, v and wwe have uFv=vFu

and u2FF(uFv) = uF(u2FFv),where u2F=

uFu, equipped with a scalar product •|• such that

(uFv)|w=v|(uFw).

In the following text, we will designate a

real ﬁnite-dimensional Euclidean Jordan algebra by

RFEJA and, we will designate an Euclidean Jordan al-

gebra only by EJA. And, the unit element of a RFEJA

or of a EJA will be denoted by e.

Example 1 Let’s consider a natural number mand

the real vector space B=Rmwith the usual vector

operations of addiction of vectors and the usual mul-

tiplication of a vector by a real scalar. Then if in Bwe

LUIS VIEIRA

University of Porto, Faculty of Engineering, Department of Civil Engineering

Street Dr Roberto Frias,0351-4200-465, Porto, PORTUGAL

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consider a multiplication Fof two of it’s elements u

and v, such that:

uFv= (u1v1, u2v2,···, umvm)

with u= (u1, u2,···, um)and v= (v1, v2,···, vm),

and if we consider the inner product •|• such that

u|v=Pm

j=1 ujvj.then B=Rmequipped with these

two operations is a RFEJA. So, we must show that for

any real numbers αand β, and for any vectors u, v

and wof Bwe have

(αu +βv)Fw=α(uFw) + β(vFw),

uFv=vFu,

u2FF(uFv) = uF(u2FFv),

(uFv)|w=v|(uFw),

where u2F=uFu. Let u, v and wbe elements of

Band let αand βbe real numbers. Considering the

notation u= (u1, . . . , um), v = (v1, . . . , vm), w =

(w1, . . . , wm),we have the following calculations.

(αu +βv)Fw= (αu1+βv1, . . . , αum+βvm)F

= (αu1w1+βv1w1,...,

, αumwm+βvmwm)

= (αu1w1, . . . , αumwm) +

+ (βv1w1, . . . , βvmwm)

=α(u1w1, . . . , umwm) +

+β(v1w1, . . . , vmwm)

=α(uFw) + β(vFw),

uFv= (u1, . . . , um)F(v1, . . . , vm)

= (u1v1, . . . , umvm)

= (v1u1, . . . , vmum)

=vFu,

u2F=uFu,

= (u1, . . . , um)F(u1, . . . , um)

= (u2

1, . . . , u2

u2FF(uFv) = (u2

1, . . . , u2

m)F

F(u1v1, . . . , umvm)

= (u2

1(u1v1), . . . , u2

m(umvm))

= (u1(u2

1v1), . . . , um(u2

mvm))

= (u1, . . . , um)F

F(u2

1v1, . . . , u2

mvm)

= (u1, . . . , um)F(u2FFv)

=uF(u2FFv),

(uFv)|w= (u1v1, . . . , umvm)|(w1, . . . , wm)

k=1

((ukvk)wk)

k=1

(vk(ukwk))

=v|(uFw).

So, we have proved that Bis an EJA. Herein, we note

that e= (1,···,1), the vector with all coordinates

equal to 1, is the unit vector of B.Indeed, for any uin

Bwe have

eFu= (1,...,1)F(u1, . . . , um)

= (u1, . . . , um)

= (u1, . . . , um)F(1,...,1)

=uFe

=u.

Example 2 Consider the real vector space Bof real

symmetric matrices of order m, provided with the vec-

tor product of two of its vectors, Fsuch that uFv=

uv+vu

2for any elements uand vof B,and with the in-

ner product •|• such that u|v=trace(uFv)for any

uand vof B.Then Bis a RFEJA and its unit is the

identity matrix of order n. We will denote the RFEJA

Bonly by Sym(m, R),from now on.

In the following text of this section let’s consider

am−dimensional real EJA Bprovided with the vector

product Fand with the inner product •|•,and being

ethe unit vector of B. Since Bis a power associa-

tive algebra, then the algebra spanned by uand eis

associative for any u∈ B.

Let wbe an element of B.The rank of

wis the smallest natural number lsuch that

{e, w1F, . . . , wlF}is a linearly dependent set of B

and we write rank(w) = l. Since dim(B) = mthen

rank(w)≤m. The rank of the RFEJA Bis the natural

number rsuch that r=rank(B) = max{rank(w) :

w∈ B}.One says that an element wof Bsuch that

rank(w) = rank(B)is a regular element of the RFEJA

B.Herein, we must say that the set of regular elements

of the RFEJA Bit is a dense set in B.

Example 3 Let’s consider the EJA B=Rmof exam-

ple 1. Then an element d= (d1, d2, . . . , dm)is an

idempotent if and only if di= 1 or di= 0,for i=

1,···, m. If d= (d1, d2,···, dm)is such that all of

it’s coordinates are not zero and di6=djfor i6=jand

1≤i, j ≤m, then the set {e, d1F, d2F,···, dm−1F}

={e, (d1, d2, . . . , dm),(d2

1, d2

2,···, d2

m),···,(dm−1

dm−1

2, . . . , dm−1

m)}is a linearly independent set of the

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vector space B,since we have:



1 1 . . . 1

d1d2. . . dm

.··· .

dm−1

1dm−1

2··· dm−1



6= 0.

But, since dim(B) = mthen the set

{e, d1F, d2F,···, dm−1F, dmF}is a linearly

dependent set of B.So, mis the smallest natural num-

ber such that the set {e, d1F, d2F,···, dm−1F, dmF}

is linearly dependent, then we conclude that

rank(d) = mand we conclude also that

rank(B) = mand we must also say that any el-

ement d= (d1, d2, . . . , dm)such that di6=djfor

i6=jand 1≤i, j ≤mis a regular element of B.

Let’s consider a regular element wof a RFEJA

Band the natural number q=rank(w).Since wqF

belongs to the real vector space spanned by the set

{e, w1F, . . . , wq−1F}then there exists real scalars

β1(w), β2(w), . . . , βq−1(w)and βq(w)such that

wqF−β1(w)wq−1F+··· + (−1)qβq(w)e= 0,(1)

being 0is the zero element of B.Taking into account

(1) we call to the polynomial psuch that

p(w, λ) = λq−β1(w)λq−1+··· + (−1)qβq(w)(2)

the characteristic polynomial of wand by construc-

tion the polynomial pis the minimal polynomial of

w. The roots of the characteristic polynomial of w

are called the eigenvalues of w. If uis not a regular

vector of Bthen we conclude that its minimal poly-

nomial has a degree less than q. And we deﬁne the

characteristic polynomial of a non regular element u

in the following way: we consider a succession unof

regular elements converging to uand next we deﬁne

p(u, λ) = limn→+∞p(un, λ),we note that the func-

tions αiare homogeneous functions of degree iover

the coordinates of uon a ﬁxed basis of B.One calls to

β1(u)the trace of uand write trace(u) = β1(u)and

we call to βq(u)the determinant of uand we write

det(u) = αq(u).

Example 4 Let’s consider the RFEJA Bof example

1 and let u= (α1, α2, . . . , αm)supposing that all

the αis are distinct. Then uis a regular element

of Band we have rank(u) = m. So, the set S=

{(1,1,···,1), u2F, . . . , um−1F}is a linearly inde-

pendent set of Band since dim(B) = mthen we con-

clude that Sis a basis of B.Now, let’s consider the

polynomial pdeﬁned through the equality (3).

p(λ) =

i=1

(λ−αi).(3)

Since p(αi)=0,∀i= 1,···, m,

then, we can write that αm

k=1(−1)k+1 P1≤i1<i2<···<ik≤mQk

l=1 αilαm−k

And, therefore we have that umF=

k=1(−1)k+1 P1≤i1<i2<···<ik≤mQk

l=1 αilum−kF.

Hence, we conclude that, umF−

k=1(−1)k+1 P1≤i1<i2<···<ik≤mQk

l=1 αilum−kF=

0m, where 0mis the zero vector of Band

u0F= (1,1,···,1).So, we conclude that the

characteristic polynomial of uis the polynomial p

deﬁned by the equality (4).

p(u, λ) = λm+

k=1

(−1)kβk(u)λm−k(4)

with β1=α1+α2+··· +αm, βr=α1,···, αm.

So, we have trace(u) = α1+α2+··· +αmand

det(u) = Qm

j=1 αj.And, we have also that βj=

P1≤i1<i2<···<ij≤mQj

l=1 αil.Next, let’s consider the

real ﬁnite Euclidean subalgebra of B,R[u] = {α1e+

α2u1F+··· +αmum−1F, αi∈R,∀i= 1,···, m}

and the linear application L(u)such that L(u)y=

uFy, ∀y∈R[u].We have that:

L(u)e=u1F= 0e+u1F+

m−1

j=2

0ujF

L(u)u1F= 0e+ 0u1F+u2F+

m−1

j=3

0ujF

L(u)um−1F= (−1)m+1βm(u)e+··· +

+ (−1)2β1(u)um−1F

So, the matrix of the linear application L(u)on

the basis S=< , u1F,···, um−1F>is the matrix:

ML(u)=







0 0 ··· 0 (−1)m+1βm(u)

1 0 ··· 0 (−1)mβm−1(u)

0 1 ··· 0 (−1)m−1βm−2(u)

.··· .

0 0 ··· 1 (−1)2β1(u)







So have that Trace(ML(u)) = β1(u)and

Det(ML(u)) = βm(u).

A vector win Bsuch that w2F=wis an idempotent

of B. Two idempotent uand vof Bare orthogonal if

uFv= 0,herein, we must say that u|v= 0 and this

happen, since we have u|v= (uFe)|v=e|(uFv) =

e|0=0.Let t∈N+ 1.The set S={d1, d2, . . . , dt}

is a complete system of orthogonal idempotent of Bif

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d2F

i=difor i= 1,···, t,uFv= 0 for u6=vand

uand vin S, and ﬁnally if Pt

j=1 dj=e. An idem-

potent is primitive if is a nonzero idempotent of Band

cannot be written as a sum of two nonzero orthogonal

idempotent. Let l∈N+ 1, the set {d1, d2, . . . , dl}is

a Jordan frame of Bif {d1, d2, . . . , dl}is a complete

system of orthogonal idempotent of Bsuch that each

idempotent difor i= 1, . . . , l is primitive.

From now on we will designate a complete sys-

tem of orthogonal idempotent and Jordan a frame, re-

spectively by CSOI and by JF. An important prop-

erty of a JF of a real ﬁnite-dimensional EJA Bis that

#B=rank(B).

Example 5 Consider the EJA of ex-

ample 1, B=R5.The set S1=

{e1, e2, e3, e4, e5}={(1,0,0,0,0),(0,1,0,0,0),

(0,0,1,0,0),(0,0,0,1,0),(0,0,0,0,1)}is a CSOI

of B.And the set S1is also a JF of B. But the set

S2={g1, g2, g3, g4}={e1, e2, e3+e4, e5}is a

CSOI of Bbut is not a JF of B, since g3is not a

primitive idempotent, indeed we have g3=e3+e4

and e3and e4are idempotent and e3Fe4= 0,where

0is the zero vector of B.

Example 6 Let’s consider the RFEJA B=

Sym(4,R).Then the set

S={d1, d2, d3, d4}

=















20 0

0 0 0 0







,





2−1

20 0

−1

20 0

0 0 0 0







=





0 0 0 0

0 0 1







,





0 0 0 0

0 0 1

2−1

0 0 −1

















is a JF of the RFEJA B.But, the set {d1+d2, d3+d4}

is a CSOI of the RFEJA B,but not a JF of B.

Theorem 7 ( [2], p. 43). Let Bbe a RFEJA. Then for

xin Bthere exist unique real numbers γ1, γ2, . . . , γt,

all distinct, and a unique CSOI {d1, d2, . . . , dt}of B

such that

x=γ1d1+γ2d2+··· +γtdt.(5)

The numbers γj’s of (5) are the eigenvalues of xand

the decomposition (5) is the ﬁrst spectral decomposi-

tion of x.

Example 8 Let d= (d1, d2, . . . , dm)be an element

of the RFEJA B=Rmof example 1 then we have that

there exists the JF {e1, e2, . . . , em}, where the vectors

eifor i= 1,···, m are the vectors of the canonical

basis of Bsuch that

d=d1e1+d2e2+···, dmem.(6)

and (6) is the second spectral decomposition of d.

Example 9 Let’s consider the RFEJA Sym(n, R).

Then if Ais a matrix with k distinct eigenvalues

λ1, λ2,···, λk−1and λkthen we have that decompo-

sition (7).

A=λ1P1+λ2P2+··· +λkPk,(7)

is the ﬁrst spectral decomposition of Awhere Pi=

j=1,j6=i(A−λjIn)

λi−λjfor i= 1,···, k.

Theorem 10 ( [2], p. 44). Let Bbe a RFEJA such

that rank(B) = rand the vector x∈ B.Then, there

exist a JF {d1, d2,···, dr−1, dr}of Band real num-

bers γ1, γ2,···, γr−1and γrsuch that we have:

x=γ1d1+γ2d2+··· +γrdr.(8)

The decomposition (8) is called the second spectral

decomposition of x.

Example 11 Let’s consider the RFEJA B=

Sym(4,R)and let’s consider the matrix A=







0 1 0 0

1 0 0 0

0 0 1 0

0 0 0 1







Then the decomposition (9) is a second spectral

decomposition of the matrix A.

A=





20 0

0 0 0 0





−





2−1

20 0

−1

20 0

0 0 0 0







+





0 0 0 0

0 0 1





−1





0 0 0 0

0 0 1

2−1

0 0 −1







(9)

and −1and 1are the eigenvalues of A. And, the ﬁrst

spectral decomposition of the matrix is the decompo-

sition presented on (10):

A=





20 0

0 0 1







+ (−1) 





2−1

20 0

−1

20 0

0 0 1

2−1

0 0 −1







.(10)

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Remark 12 Let’s consider the RFEJA B=

Sym(m, R). a matrix Awith the distinct eigenvalues

λ1, λ2,···, λk,and let’s consider a orthonormal ba-

sis of Rn, S =< v1, v2,···, vm>where we suppose

each viis a column vector of Rmfor i= 1,···, m.

Then, considering Ci=vivT

ifor i= 1,···, m. then

we obtain the ﬁrst spectral decomposition of Ais

presented on inequality (11).

j=1

λijCl(11)

Where the eigenvalues λij∈ {λ1, λ2,···, λk}.

Remark 13 Let’s consider the real ﬁnite Euclidean

Jordan algebras B1and B2with the operations of

multiplication F1and F2respectively, and with the

inner products •|1•and •|2•respectively. Then,

(B1;B2)equipped with the vector multiplication of

vectors (a;b)F(c;d) = (aF1c;bF2d)and the inner

product (a;b)|(c;d) = a|1c+b|2dis a RFEJA. And,

if e1and e2are the units of B1and of B2respectively,

then (e1;e2)is the unit of RFEJA (B1;B2).

A Euclidean Jordan algebra is simple if it does not

contain any non trivial ideal.

We are concerned on those RFEJA Bsuch that

rank(B) = dim(B) = m, since in those EJA any

Jordan frame S={d1, d2, . . . , dm}is a basis of the

Euclidean Jordan algebra Band we can deﬁne ||x||

for x=λ1d1+λ2d2+. . . +λmdmby the equal-

ity ||x|| =ptrace(x◦x) = qPm

i=1 λ2

i.Herein we

must note that if uis a primitive idempotent then

trace(u)=1.

3 Some concepts about strongly reg-

ular graphs

In this section, we present the more important con-

cepts about strongly regular graphs and we follow the

text presented in the paper [1]. The works, see [18],

“ A course in Combinatorics” and “Algebraic Graph

Theory” , see [19], are very good on the description

of the algebraic properties of a strongly regular graph.

A graph Gis a pair of sets {V(G), E(G)}such

that V(G) = {v1, v2,···, vm}is the set of vertexes,

nodes or points of Gand E(G)is the set of edges of

G. Is natural to represent an edge between the ver-

texes vkand vlby vkvl.A graph Gis called simple

if it has non-parallel edges or loops. The number of

vertexes of a graph Gis called the order of Gand the

dimension of Gis the number of edges of G.

A graph Gof order mis called a null graph if

E(G) = ∅and V(G)6=∅.And, a simple graph Gof

order mis a complete graph if all pair of distinct ver-

texes of V(G)are adjacent vertexes. And one denotes

the complete graph of order mby Km.

The complement graph of a simple graph Gthat

is denoted by Gis a simple graph with the same set of

vertexes of Gand such that two any of its vertexes are

adjacent if and only if they are not adjacent vertexes

of G.

From now on, we only consider non-empty, sim-

ple and non-complete graphs. One says that an edge of

E(G)is incident on a vertex v∈V(G)if and only if v

is an extreme vertex of this edge. The extreme points

of an edge of the graph Gare called adjacent vertexes

or neighbors. The set of vertexes that are neighboring

vertexes of a vertex vis called the neighborhood of

the vertex v, one denotes this set by NG(v).

One deﬁnes the degree of a vertex vof a graph

Gthe number of incident vertexes on v. A graph G

is called l−regular if all of its vertexes have the same

degree l.

A graph G, is called a (m, l;c, d)-strongly regu-

lar graph if Gis as graph of order m, is a l−regular

graph such that any pair of adjacent vertexes have c

common neighbor vertexes and any pair of non adja-

cent vertexes have dcommon neighbor vertexes. In

the following text of this section, we will designate a

strongly regular graph by srg.

If Gis a (m, l;c, d)−srg then the complement

graph of G, G is a (m, m −l−1; m−2l+d−2, m −

2l+c)−srg.

A(m, l;c, d)−srg Gis primitive if and only if G

and Gare connected. A (m, l;c, d)−srg is a non prim-

itive srg if and only if d=lor d= 0.

Consider a (m, l;c, d)−srg Gand BGits adja-

cency matrix. Then BG= [bij ], where BGis a matrix

of order msuch that bij = 1, if the vertex iis adja-

cent to jand 0otherwise. The adjacency matrix of G

satisﬁes the equation (12).

G=l Im+cBG+d(Jm−BG−Im).(12)

One deﬁnes the eigenvalues of Gas being the eigen-

values of BG.The eigenvalues of G, are l, θ and τ,

see [19], where

θ= (c−d+p(c−d)2+ 4(l−d))/2,(13)

τ= (c−d−p(c−d)2+ 4(l−d))/2.(14)

Next, we refer to some feasibility conditions over

the natural numbers c, d, l and mfor the existence of

a(m, l;c, d)primitive srg G.

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The multiplicities fθand fτare deﬁned by the

inequalities (15) and (16).

fθ=|τ|n+z−k

θ−τ,(15)

fτ=θn +k−θ

θ−τ.(16)

The conditions fθ∈Nand fτ∈Nare called the

integrability conditions of a srg. Next we describe the

called Krein conditions (17) and (18)

(τ+ 1)(l+τ+ 2θτ)≤(l+τ)(θ+ 1)2,(17)

(θ+ 1)(l+θ+ 2θτ)≤(l+θ)(τ+ 1)2,(18)

that have being deduced in the article [20].

Example 14 Let’s consider the parameter set

(m, l;c, d) = (56,22; 3,12).If this sequence would

correspond to a strongly regular graph then we would

have θ= 1 and τ=−10.But, then we would have:

(τ+ 1)(1 + τ+ 2θτ) = 261,(19)

(l+τ)(θ+ 1)2=−36.(20)

Hence, these values of θand τviolate the Krein con-

dition (17) and therefore doesn’t exist any srg with the

parameters (56,22; 3,12).We must say that the Krein

condition (18) is not violated by the values of θ= 1

and τ=−10.

Next, we present the equality (21) that the parameters

m, l, c and dof a (m, l;c, d)−primitive srg satisﬁes .

l(l−1−c) = d(m−l−1).(21)

And, ﬁnally, we must say that if we consider the

(m, l;c, d)−primitive srg then the multiplicities of the

eigenvalues τand θmust verify the inequalities (22)

and (23).

n≤fθ(fθ+ 3)

2,(22)

n≤fτ(fτ+ 3)

2.(23)

The admissibility conditions (22) and (23) are known

as the absolute bound feasibility conditions for the ex-

istence of a strongly regular graph.

Example 15 Let’s consider the parameter set

(m, l;c, d) = (64,30; 18,10).Then, recurring to

(13) and (14) we get θ= 10 and τ=−2.Using the

equalities (15) and (16) we have:

fθ= 8

fτ= 55.

Since

n= 64

fθ(fθ+ 3)

2= 44

then the absolute bound condition (22) is not satis-

ﬁed therefore don’t exist any strongly regular corre-

sponding the parameter set (64,30; 18,10).We must

say that the absolute bound condition (23) is not vio-

lated.

4 Some inequalities over the eigen-

values of a strongly regular graph

In this section we will establish some inequalities over

the parameters and the spectrum of a primitive srg.

But ﬁrstly, we present some notation for the Schur

product of matrices.

Given two real square matrices Aand Bof order

none considers the Schur product ◦of these two ma-

trices as being the matrix C=A◦Bsuch that consid-

ering the notation C= [cij ], A = [aij ]and B= [bij ]

then cij =aij bij .And, one deﬁnes the Schur powers

of a square matrix Afor a natural number jas being

the matrix Aj◦such that A◦0=Jn, A1◦=Aand

Aj◦=AAj−1◦for j≥2.

Let’s consider the (m, l;c, d)−primitive strongly

regular graph Gwith the adjacency matrix A, such

that 0< d < l−1.Firstly, we will suppose that l < m

and that c > d. Next, let’s Abe the Euclidean Jordan

subalgebra Aof the RFEJA Sym(m, R)spanned by

the identity matrix of order mand the powers Ai, i ∈

N.We have that rank(A) = dim(A)=3.The eigen-

values of the matrix Aare , λ1=c−d+√(c−d)2+4(l−d)

and λ2=c−d−√(c−d)2+4(l−d)

2and B={F1, F2, F3}

is a JF of Awhere:

F1=1

mIm+1

mA+1

m(Jm−A−Im) = Jm

F2=|λ2|m+λ2−l

n(λ1−λ2)In+m+λ2−l

m(λ1−λ2)A+

+λ2−l

m(λ1−λ2)(Jm−A−Im),

F3=λ1m+l−λ

m(λ1−λ2)Im+−m+l−λ1

m(λ1−λ2)A+

+l−λ1

m(λ1−λ2)(Jm−A−Im),

where Jmis the real symmetric matrix of order m

where all entries are the real number 1and Imis the

identity matrix of order m.

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Let be a real positive number. Since the Schur

power series P+∞

k=0 1

l!ln(1 + )A2

l+k◦

is convergent

then let Sbe its sum. Using he fact that A2=lIm+

cA +d(Jm−A−Im)then we conclude that:

+∞

i=0

i!ln(1 + )l

l+i

Im+

+∞

i=0

i!ln(1 + )c

l+i

A+

+∞

i=0

i!ln(1 + )d

l+i

(Jm−A−Im).

Hence, we obtain the equality (24).

S= (1 + )l

l+In+ (1 + )c

l+A+

+ (1 + )d

l+(Jm−A−Im).(24)

Let’s consider the spectral decomposition of the par-

tial sum of order n, Sn=Pn

i=0 1

i!ln(1 + )A2

l+i◦,

Sn=qn1F1+qn2F2+qn3F3.Since the spectral de-

composition S=q1F1+q2F2+q3F3is such that

qi= limn→+∞qni for i= 1,···,3and qni ≥0for

i= 1,···,3then the qis are all non negative. Next,

let’s consider the element F3◦Sof A.So, we can

write the equality (25).

F3◦S=λ1m+l−λ1

m(λ1−λ2)(1 + )l

l+Im+

+−m+l−λ1

m(λ1−λ2)(1 + )c

l+A+

+l−λ1

m(λ1−λ2)(1 + )d

l+(Jm−A−Im).

(25)

Recurring, to the spectral decomposition of F3◦S=

q31F1+q32F2+q33F3we conclude that

q31 =λ1m+l−λ1

m(λ1−λ2)(1 + )l

l++

+−m+l−λ1

m(λ1−λ2)(1 + )c

l+l+

+l−λ1

m(λ1−λ2)(1 + )d

l+(m−l−1)(26)

since we have the equality (27),

λ1m+l−λ1

m(λ1−λ2)+−m+l−λ1

m(λ1−λ2)l+

+l−λ1

m(λ1−λ2)(m−l−1) = 0

(27)

then, after some algebraic manipulation of the expres-

sion of q31 recurring to (27) we obtain the equality

(28).

q31 =λ1m+l−λ1

m(λ1−λ2)(1 + )l

l+−(1 + )d

l+

−m−l+λ1

m(λ1−λ2)(1 + )c

l+−(1 + )d

l+l.

(28)

So, since q31 ≥0then we obtain the inequality

(29).

λ1m+l−λ1

m(λ1−λ2)(1 + )l

l+−(1 + )d

l+

≥m−l+λ1

m(λ1−λ2)(1 + )c

l+−(1 + )d

l+l.

(29)

Next, applying the Lagrange Theorem to the function

fsuch that f(x) = (1 + )x,∀x∈Ron the inter-

vals [d

l+,l

l+]and [d

l+,c

l+], and majoring and mi-

noring the function fon those intervals we obtain the

inequality (30),

λ1m+l−λ1

m(λ1−λ2)(1 + )l

l+l−d

l+

≥m−l+λ1

m(λ1−λ2)(1 + )c

l+c−d

l+l(30)

this is we obtain the inequality (31).

λ1m+l−λ1

m(λ1−λ2)(l−d)

≥m−l+λ1

m(λ1−λ2)(c−d)l. (31)

and, ﬁnally we get (32).

λ1m+l−λ1

m(λ1−λ2)≥m−l+λ1

m(λ1−λ2)

c−d

l−dl. (32)

Next, supposing that l < m

3,rewriting (32) and after

some algebraic manipulation of (32) we deduce (33)

3λ1+ 1 ≥2c−d

l−dl. (33)

And, ﬁnally, since λ1>1we conclude that the in-

equality (34) is veriﬁed.

4λ1≥2(c−d)l

l−d(34)

and, so we have

λ1≥(c−d)l

2(l−d).

Then, we have established the Theorem 16.

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Theorem 16 Let’s consider the (m,l;c,d)−prim-

itive srg Uof order msuch that 0< d < l −1,c > d

and l < m

3.Then we the equality (35) is veriﬁed.

λ1≥(c−d)l

2(l−d).(35)

where λ1is the positive eigenvalue of Udistinct from

the regularity of U.

Next, we will construct new inequalities over the

parameters of a strongly regular graph Grecurring to

its Generalized Krein parameters. As, is known the

Generalized Krein parameters of a strongly regular

graph G,qijl;mn are deﬁned, see [15], as being the

real numbers such that:

Fm◦

i◦Fn◦

l=1

qijl;mnFl

where i, j, l ∈ {1,2,3}and mand nare natural num-

bers such that at least one of them is greater than

1.And, the Generalized Krein parameters qil;m,with

i, l, m natural numbers such that 1≤i, l ≤3and

m≥3,are the unique real numbers such that:

Fm◦

l=1

qil;mFl.

Next, let’s suppose that l < m

3.And, let’s analyze

the Generalized Krein parameter q31;3.Firstly, since

this parameters is non negative we have the following

inequality (36).

λ1m+l−λ1

m(λ1−λ2)3

+−m+l−λ1

m(λ1−λ2)3

+l−λ1)

m(λ1−λ2)3

(m−l−1) ≥0.(36)

Since the equality (37) is veriﬁed

λ1m+l−λ1

m(λ1−λ2)+−m+l−λ1

m(λ1−λ2)l+

+l−λ1

m(λ1−λ2)(m−l−1) = 0,(37)

after some algebraic manipulation of (36)

and using the equality (37) we conclude that:

λ1m+l−λ1

m(λ1−λ2)(λ1m+l−λ1)2

m2(λ1−λ2)2−(l−λ1)2

m2(λ1−λ2)2≥m−l+λ1

m(λ1−λ2)

(m−l+λ1)2

m2(λ1−λ2)2−(l−λ1)2

m2(λ1−λ2)2lthis is we deduce that:

λ1m+l−λ1

m(λ1−λ2)λ1m+2l−2λ1

m(λ1−λ2)  λ1

λ1−λ2≥m−l+λ1

m(λ1−λ2)

m−2l+2λ1

m(λ1−λ2)1

λ1−λ2l. Next, supposing that l < m

3we

conclude that (38) is veriﬁed.

(3λ1+ 1)(3λ1+ 2)λ1≥2l. (38)

If c > d then we conclude that λ1>1and therefore

noting the following writing of (38),

λ2

1(3λ1+ 1

λ1

)(3λ1+ 2

λ1

)λ1≥2l(39)

we conclude from (39) that (40) is veriﬁed.

20λ3

1≥2l, (40)

and therefore in this case we have established the in-

equality (41).

λ3

1≥l

10.(41)

Hence, we have established the Theorem 17.

Theorem 17 Let’s consider the (m, l;c, d)−primi-

tive srg Uof order msuch that 0< d < l −1and

c > d, l < m

3then the equality (42) is veriﬁed.

λ3

1≥l

10,(42)

where λ1is the positive eigenvalue of Udistinct from

the regularity of U.

5 Conclusion

The results obtained in this paper are distinct from

those obtained in the publication [1] and these in-

equalities over the eigenvalues of a primitive strongly

regular graph are obtained recurring to methods dis-

tinct of those used on the paper [1]. On the future di-

rection of research we will use other methods of spec-

tral analysis to establish more general feasibility con-

ditions for the existence of a strongly regular graph.

Acknowledgements: Lu´

ıs Vieira in this work

was partially supported by the Center of Re-

search of Mathematics of University of Porto

(UID/MAT/00144/2013), which is funded by FCT

(Portugal) with national (MEC) and European struc-

tural funds through the programs FEDER, under the

partnership agreement PT2020.

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References:

[1] L. Vieira, Euclidean Jordan algebras and some

properties of strongly regular graphs, Proceed-

ings 2022 7th International Conference on Math-

ematics and Computers in Sciences and Industry,

MCSI 2022, pp. 18-23.

[2] J. Faraut and A. Kor´

anyi, Analysis on Symmetric

Cones, Science Publications, Oxford, 1994.

[3] V. Mano and L. A. Vieira, Strongly Regular

Graphs and Euclidean Jordan Algebras Revela-

tions within an Unusual Relationship, Academic

Lambert Publishing, Berlin, 2015.

[4] M. S. Gowda, Some majorization inequalities

induced by Schur products in Euclidean Jordan

Algebras, Linear Algebra Appl, Vol.600, 2020,

pp.1-21.

[5] M. S. Gowda, Simultaneous spectral decompo-

sition in Euclidean Jordan algebras and related

systems,Linear&Multilinear Algebra, Vol.70,

2022, pp. 6535-6547.

[6] J. Tao, An analog of Thompson’s triangle in-

equality on Euclidean Jordan algebras, Eletronic

Journal of Linear Algebra, Vol.37, 2021,

pp. 156-159.

[7] L. Faybusovich, A Jordan algebraic approach to

potential reduction algorithms, Mathematische

Zeitschrift, Vol.239, 2002, pp. 117–129.

[8] M. S. Gowda, J. Tao and M. Moldovan, Some

inertia Theorems in Euclidean Jordan algebras,

Linear algebra and its applications, Vol.430,

2009, pp. 1992–2011.

[9] D. M. Cardoso and L. A. Vieira, On the opti-

mal parameter of a self-concordant barrier over

a symmetric cone, European Journal of Opera-

tional research, Vol.169, 2006, pp. 1148–1157.

[10] F. Alizadeh and D. Goldfard, Second Order

Cone Programming, Math. Programming Series

B, Vol. 95, 2004, pp. 3–51.

[11] M. M. Moldovan and M. S. Gowda, Strictly di-

agonal dominance and a Gersgorin type theo-

rem on euclidean Jordan algebras, Linear Alg.

Appl. 431, 2009, pp.148-161.

[12] S. H. Schmieta and F. Alizadeh, Extension of

primal-dual interior point algorithms to symmet-

ric cones, Math. Prog. Series A, Vol. 96, 2003,

pp. 409-438.

[13] D. Sossa, A Fiedler type determinant inequal-

ity in Euclidean Jordan algebras, Linear Alge-

bra and its Applications, Vol.667, 2023, pp.151–

164.

[14] A. Seeger, Condition Number minimization in

Euclidean Jordan algebras, Siam Journal on Op-

timization, Vol.32, 2022, pp. 635–658.

[15] L. A. Vieira, Generalized inequalities associ-

ated to the regularity of a strongly regular graph,

Journal of Computational Methods in Sciences

and Engineering, No.19, No.3, 2019, pp. 673–

680.

[16] L. A. Vieira, Euclidean Jordan algebras, strongly

regular graphs and Cauchy Schwarz inequalities,

Appl. Math, Vol.13, No.3, 2019, pp. 437–444.

[17] F. Alizadeh and S. H. Schmieta, Symmetric

Cones, Potential Reduction Methods and Word

by Word Extensions, Handbook of Semideﬁnite

Programming, Theory, Algorithms and Appli-

cations (R. Saigal, L. Vandenberghe, and H.

Wolkowicz, ed.), Kluwer Academic Publishers,

2000, pp. 195-233.

[18] J. H.V. Lint and R. M. Wilson, A Course in Com-

binatorics, Cambridge University Press, Cam-

bridge University Press, Cambridge, 2001.

[19] C. Godsil and G. F. Royle, Algebraic Graph The-

ory, Springer Verlag, New York, 2001.

[20] Jr. L. L. Scott, A condition on Higman’s param-

eters, Notices of Amer. Math. Soc, Vol.20, 1973,

A-97.

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