Euclidean Jordan algebras with strongly regular

graphs

Domingos M. Cardoso1 (dcardoso@mat.ua.up.pt)Luıs Vieira2 (lvieira@fe.up.pt)

1 Universidade de Aveiro, Departamento de Matematica,3810-193 Aveiro - Portugal.

2 Universidade do Porto, Faculdade de Engenharia,Seccao de Matematica e Fısica do Dep. de Eng. Civil,

4200-465 Porto - Portugal.

February 14, 2003

Abstract

An Euclidean Jordan algebra Vn of dimension m = bn2c + 1, for ar-

bitrary n, is produced. The algebraic and combinatorial properties of abasis of Vn allows the determination of strongly regular graphs of ordern, when n is even. The character table of this Euclidean Jordan algebrais deduced and applied on the spectra recognition of the above stronglyregular graphs included in Vn.

Keywords: Euclidean Jordan Algebras; Graph Theory (17C27; 05C50).

1 Introduction

Homogeneous coherent symmetric algebras, also known as symmetric associa-tion schemes [1], are particular cases of cellular algebras [9] and also particularcases of Euclidean Jordan algebras [8]. Cellular algebras appear on the studyof symmetry and isomorphism of graphs, using matrix stabilization procedures,and also in the framework of the automorphism groups of graphs [4] and [7].Among several readable texts about or including Euclidean Jordan algebras, wemay refer [3] and [8]. Previous work on strongly regular graphs (see for instance,[5] and [9]) were developed aiming to obtain the minimal coherent algebra whichincludes a particular set of matrices and then its symmetric homogeneous co-herent 3-dimension subalgebras, all of them defining strongly regular graphs [6].In this paper, from a particular element in a basis B = {A1, · · · , Am} (whereA1 = In is the identity matrix of order n) of an Euclidean Jordan algebra Vn,with as many different eigenvalues as the dimension of the algebra, another ba-sis B′ of idempotents is obtained by exploiting the algebraic and combinatorialproperties of Vn. From B′, we get easily the character table of Vn. Then, ina way similar to the one used in [2] but in the different context of associationschemes, fusing all the matrices of B but A1 and Am, a strongly regular graph

1

is obtained when n is even. Additionally, for particular even values of n, otherstrongly regular graphs included in the Euclidean Jordan algebra Vn are de-duced.The notation will be introduced throughout the remaining sections which are or-ganized as follows. In the section 2 we present the basics about Euclidean Jordanalgebras and the Euclidean Jordan algebra Vn which has particular combina-torial properties. In section 3 we introduce a procedure for the determinationof the character table of Vn. In section 4 some strongly regular graphs whoseadjacency matrices belong to Vn are deduced and the corresponding spectradetermined. Finally, in section 5 a numerical example is described.

2 An Euclidean Jordan algebra with particularcombinatorial properties

An abstract Euclidean Jordan algebra is herein denoted by V and the EuclideanJordan algebra on the vector space of real matrices of order n by Vn.Let V be a finite m-dimensional Euclidean vector space over the real numbers,where the inner product is < .|. > . If exists a bilinear application ◦ : V ×V → Vsuch that ∀x, y ∈ V

x ◦ y = y ◦ x, (1)∃e ∈ V : x ◦ e = x, (2)

x ◦ (x2 ◦ y) = x2 ◦ (x ◦ y), (3)< (x ◦ y)|z > = < y|(x ◦ z) >, (4)

then V is an Euclidean Jordan algebra with unit element e. As it will beshown later on, imposing few conditions on V there is a basis < c1, · · · , cm > ofidempotents (that is, such that c2

i = ci) pairwise orthogonal (that is, ci ◦ cj =0 ∀i 6= j) and such that

m∑i=1

ci = e. (5)

Usually, a set of orthogonal idempotents {c1, · · · , cs} of V, for which (5) holdsis called a complete system of orthogonal idempotents.

From property (4) it follows that

∀x, y ∈ V x ◦ y = 0 ⇒< x|y >= 0.

Therefore if {c1, · · · , cs} is a complete system of orthogonal idempotents then{c1, · · · , cs} is a linear independent set of V. So, every complete system of or-thogonal idempotents of V with cardinality m is a basis for V.The rank of an element x ∈ V is denoted by rank(x) and defined as

rank(x) = min{s ∈ N : {e, x · · · , xs} is linear dependent}.

Since V has dimension m, we may conclude that ∀x ∈ V rank(x) ≤ m. The rankof the Euclidean Jordan algebra V is defined as rank(V) = max{rank(x) : x ∈V}. The elements of V such that rank(x) = rank(V) are usually called regular

2

elements of V. As it will be proved latter on, if exists a regular element x ∈ Vsuch that rank(x) = dim(V) (and therefore rank(V) = dim(V)), then there ex-ists a basis of V which is a complete system of orthogonal idempotents.

For each x ∈ V, L(x) denotes the linear application

L(x) : V 7→ Vy L(x)y = x ◦ y.

Furthermore R[x] denotes the subspace of V spanned by e and the naturalpowers of x. Let us denote the restriction of L(x) to R[x] by L0(x) and let usconsider the linear application L : V 7→ End(V), where End(V) denotes the setof endomorphisms of V, such that L(x) = L(x). Since, for x ∈ V, if L(x) = O(where O denotes the null operator) then 0 = O(e) = L(x)e = x ◦ e ⇒ x = 0and hence L is injective.Let r be a natural number such that < e, x, · · · , xr > is a basis of R[x] and letu, v ∈ R[x]. Then

u =r∑

i=0

αiuxi and v =r∑

i=0

αivxi,

where x0 = e. From (4) follows that

< L0(x)xs|xt >=< xs|L0(x)xt > ∀s, t ∈ {0, · · · , r}.

Therefore ∀u, v ∈ R[x] < L0(x)u|v >=< u|L0(x)v > and we may conclude thatthe linear operator L0(x) is self adjoint. Thus L0(x) is a symmetric operatorand then it has only real eigenvalues. Denoting by λ1, · · · , λk the k differenteigenvalues of L0(x) and by V1, · · · , Vk the corresponding eigenspaces, it followsthat ∀i ∈ {1, · · · , k} the projector on each subspace Vi is

Pi = Πi 6=jL0(x)− λjI

λi − λj. (6)

Indeed, from the definition (6), it is immediate to conclude the projector prop-erties

(i)∑k

i=1 Pi = In

and

(ii) PiPj = δijPi ∀i, j ∈ {1, · · · , k}, where δij denotes the Kronecker symbol,which is such that δij = 1 if i = j and δij = 0 otherwise.

Therefore, considering

ci = Πi 6=jL0(x)− λjI

λi − λj(e)

= Πi 6=jx− λje

λi − λj. (7)

for i = 1, · · · , k, and taking into account that the linear operator L is injective,we may conclude that c1, · · · , ck is a complete system of orthogonal idempotentssuch that

x =k∑

i=1

λici. (8)

3

The equality (8) is known as the spectral decomposition of x and the scalarsλi are called the eigenvalues of x. If x ∈ V has m different eigenvalues λ1, · · · , λm,then we have a basis B′ of V whose elements are defined by (7).Suppose that the Euclidean Jordan algebra V includes an element x such thatL0(x) has m different eigenvalues and that the spectral decomposition of x isx =

∑mi=1 λici. Then, aiming to obtain the spectral decomposition of any el-

ement y ∈ V, all we need is to determine the coordinates of y on the basis< c1, · · · , cm > . Furthermore, since the goal is to determine strongly regulargraphs, is better to deal with an Euclidean Jordan algebra Vn which is alsoa homogeneous symmetric coherent algebra. The algebraic and combinatorialproperties of V will be also exploited, in order to obtain a general formula forthe spectral decomposition of any of its elements.

Consider the matrices Eij ∈ Rn×n ∀i, j ∈ {1, · · · , n}, defined by (Eij)pq =δipδjq. Let n be a natural number, m = bn

2 c + 1 and the family of matricesF = {Ai}i∈{1,···,m}, with A1 = In,

Ar =n∑

l=r

El, l−r+1 +n∑

l=r

El−r+1, l +r−1∑l=1

En−r+1+l, l +r−1∑l=1

El, n−r+1+l,

for r = 2, · · · ,m− 1, and

Am =n∑

l=m

El, l−m+1 +n∑

l=m

El−m+1, l +m−1∑l=1

En−m+1+l, l +m−1∑l=1

El, n−m+1+l

if n is odd, and

Am =m−1∑l=1

En−m+1+l, l +m−1∑l=1

El, n−m+1+l

if n is even.

From the definition, it is clear that the matrices Aj , for j = 2, . . . ,m, are allsymmetric, with 0− 1 entries and with the diagonal elements zero valued.

Let < Vn,+, ., < .|. >> be the Euclidean space spanned by < A1, A2, · · · , Am >over R, where +, . are the usual sum and product of matrices, respectively, and< .|. > is the inner product defined by < X|Y >= tr(XY ) (where tr(Z) denotesthe trace of Z, that is, the sum of the eigenvalues of Z).The next theorem states several usuful algebraic properties of the family F .Before that, however, it is convenient to introduce the following notation.For i = 0, · · · , n− 1 the matrices Ci are defined by

(Ci)pq ={

1 if q = p⊕n i0 if q 6= p⊕n i

∀p, q ∈ {1, · · · , n},

where ⊕n denotes the sum module n. These matrices are known as circulantmatrices. For instance, the circulant matrices obtained for n = 4, beyond theidentity matrix C0 = I4, are the following.

4

C1 =

0 1 0 00 0 1 00 0 0 11 0 0 0

, C2 =

0 0 1 00 0 0 11 0 0 00 1 0 0

and C3 =

0 0 0 11 0 0 00 1 0 00 0 1 0

.

Therefore from the definition of the matrices Ai, it follows that

A1 = C0, (9)Ai = Ci−1 + Cn−(i−1), for i = 2, . . . ,m− 1, (10)

Am = Cm−1 if m is even , (11)Am = Cm−1 + Cn−(m−1) if m is odd . (12)

Theorem 2.1The matrices of the family F verify

•

AiAj = Ai+j−1 + Aj−i+1 if{

2 ≤ i < j < mi + j ≤ m

(13)

•

AiAj = An−(i+j−1)+1 + Aj−i+1 if{

2 ≤ i < j < mi + j ≥ m + 2 (14)

•

AiAi = A2i−1 + 2In if 2 ≤ i ≤ bm + 12

c (15)

•

AiAi = An−2(i−1)+1 + 2In if i ≥ dm2e+ 1 (16)

• If n is odd then

AiAj = Ai+j−1 + Aj−i+1 if{

2 ≤ i < j < mi + j = m + 1 (17)

AiAm = An−(i+m−1)+1 + Am−i+1 if 2 ≤ i ≤ m− 1 (18)

A2m = A3 + 2In (19)

• If n is even then

AiAj = 2Am + Aj−i+1 if{

2 ≤ i < j < mi + j = m + 1 (20)

AiAm = Am−i+1 if i < m (21)A2

m = In (22)

5

Proof.The proof of the items are all similar, so we shall prove only (13). Let i, j be

natural positive numbers, such that 2 ≤ i < m and i + j ≤ m.Since the matrices Ci verify the following equalities

CiCj = Ci⊕nj ∀i, j ∈ {0, · · · , n− 1}

then

AiAj = (Ci−1 + Cn−(i−1))(Cj−1 + Cn−(j−1)),

= Ci+j−2 + Cn−(i+j−2) + Cn−(j−i) + Cj−i.

Taking into account that i + j − 1, j − i + 1 ∈ {2, · · · , m}, it follows that

AiAj = Ai+j−1 + Aj−i+1.

It follows some properties of Vn.

Theorem 2.2 Let Vn be the algebra spanned by F over R. Then Vn

(i) is commutative,

(ii) is an Euclidean Jordan algebra,

(iii) and has dimension m = rank(A2).

Proof.

(i) Since the matrices of the family F belong to the space spanned by < C0, · · · , Cn−1 >,it follows that AiAj = AjAi ∀i, j ∈ {1, · · · , m}, and then it is immediate toconclude that Vn is commutative.

(ii) Since, by (i), (Vn, .) is commutative, in order to conclude that Vn is an EuclideanJordan Algebra, it is sufficient to prove that

(a) A(A2)B = A2(AB) ∀A, B ∈ Vn.

(b) < L(A)B, C >=< B, L(A)C > ∀A, B, C ∈ Vn.

Let A, B, C ∈ V.

(a) Since (Vn, .) is a commutative and associative algebra then it follows that

A(A2)B = (AA2)B = (A2A)B = A2(AB).

(b) On the other hand,

< L(A)B, C >= tr((A.B).C) = tr((B.A).C) = tr(B, (A.C)) =< B, L(A)C > .

(iii) It is immediate that rank(A2) ≤ m. Therefore, rank(A2) = m if and only if{A1, A2, · · · , (A2)

m−1} is linear independent.From the properties of matrices Ci, for i = 0, · · · , n−1, and taking into accountthat A2 = C1 + Cn−1, using the Newton binomial formula, it follows that

(A2)2l =

l∑i=0

(2l

i + l

)C2i +

l∑i=1

(2l

i + l

)Cn−2i, (23)

for l = 1, · · · , bm

2c,

(A2)2l−1 =

l−1∑i=0

(2l − 1

l − i− 1

)C2i+1 +

l∑i=1

(2l − 1

l + i− 1

)Cn−2i+1, (24)

for l = 2, · · · , bm + 1

2c.

6

Since the matrices C0, · · · , Cn−1 are linear independent, from (23) - (24) followsthat {A1, A2, (A2)

2, · · · , (A2)m−1} is a linear independent set of Vn and then

rank(A2) = m.

Let us prove now that the minimal polynomial of A2 is

pm(X) = Πm−1i=0 (X − 2 cos(

2πi

n)In).

Since (C1)i = Ci ∀i ∈ {1, · · · , n}, then the minimal polynomial 1 of C1 ispn(X) = Xn − In. Using the relations A2 = C1 + (C1)T , C1(C1)T = (C1)T C1

and taking into account that A2 is a symmetric matrix with minimal polynomialdegree m, then the eigenvalues of A2 are λ(i+1)2 = 2 cos( 2πi

n ), for i = 0, · · · ,m−1.So, we are able to determine the set of idempotents whose elements, accordingto (6), are the projectors:

Pi = Πi 6=jL0(A2)− λj2I

λi2 − λj2(In)

= Πi 6=jA2 − λj2In

λi2 − λj2

for i = 1, · · · ,m.

Therefore B′ =< P1, · · · , Pm > is a basis of Vn and then ∀j ∈ {1, · · · ,m} ∃λij

for i = 1, · · · ,m such that

Aj =m∑

i=1

λijPi. (25)

Multiplying both member of (25) by Pi on the right, it follows that AjPi = λijPi

and then an easy way for the computation of the coordinates of Ai relatively toB′ may be deduced. On the other and, taking into account that the scalars λij ,for j = 1, · · · ,m, are the eigenvalues of Aj , the character table for Vn is the onerepresented by table 1.

A1 A2 · · · Am

P1 1 λ12 · · · λ1m

......

......

...Pm 1 λm2 · · · λmm

Table 1: The character table of Vn.

3 A procedure for the determination of the char-acter table of Vn

From the above analysis, a procedure for the determination of the charactertable of the Euclidean Jordan algebra Vn is now deduced. Consider the basis

1Note that {C0, · · · , Cn−1} is a linear independent set on Rn×n and Cn1 = In.

7

< A1, · · · , Am > . If i ∈ {1, · · · ,m} then there are scalars βki such that

Pi =m∑

k=1

βkiAk. (26)

From A2Pi = λi2Pi follows that A2Pi =∑m

k=1 λi2βkiAk. On the other hand,multiplying both members of (26) on the left by A2, it follows that A2Pi =∑m

k=1 βkiA2Ak. Therefore

m∑k=1

λi2βkiAk =m∑

k=1

βkiA2Ak. (27)

• Let us suppose that n is even. Taking into account (10)-(12),

Ai = Ci−1 + Cn−(i−1), for i = 2, · · · ,m− 2,

Am−1 = Cm−2 + Cn−(m−2),

Am = Cm−1,

and theorem 2.1, it follows that

A2A1 = A2,

A2A2 = A3 + 2In,

A2Ai = Ai+1 + Ai−1, for i = 3, · · · ,m− 2,

A2Am−1 = 2Am + Am−2,

A2Am = Am−1.

Replacing in (27) the products A2Ak, for k = 1, · · · ,m, by the left handsides of the above equalities, we get 2β2i = λi2β1i,

β(j−1)i + β(j+1)i = λi2βji, for j = 2, · · · ,m− 1,2β(m−1)i = λi2βmi.

(28)

The system (28) allows the recursively determination of the scalars βji

from the values βki, with 1 ≤ k < j, and then each coordinate of theprojector Pi may be obtained as functions of β1i. Without loss of generalitywe may assume that β1i 6= 0 2. Consider the notation

Pi = β1i

m∑k=1

βkiAk, (29)

where βki = βki

β1i, for k = 1, · · · ,m. Therefore, it is clear that β1i = 1 and

also that all the remaining βki can be obtained, recursively, from (28). Thedetermination of β1i follows from the fact that Pi is idempotent. Thencomputing

(Pi)2 = (m∑

k=1

βkiAk)2 = β21i(

m∑k=1

βkiAk)2

2Since 0 < tr(Pi) = β1itr(In) +∑m

k=2 βkitr(Ak) and tr(Ak) = 0, for k = 2, · · · , m, followsthat β1i > 0.

8

and expanding (∑m

k=1 βkiAk)2 we get

(m∑

k=1

βkiAk)2 = (β21i +

m−1∑k=2

2β2ki + β2

mi)In + · · ·

and then, since (Pi)2 = Pi, it comes that

β1i =β1i

β21i +

∑m−1k=2 2β2

ki + β2mi

. (30)

Considering the equalities (25), that is, Aj =∑m

k=1 λkjPk for each j ∈{2, · · · ,m− 1}, we get

AjPi = λijPi. (31)

Now we are able to compute the values λij as functions of the coordinatesof the projector Pi on the basis B.Consider Pi = β1iA1 +

∑mk=2 βkiAk. Noting that, from theorem 2.1, ∀k ∈

{1, · · · ,m} AjAk =∑m

r=1 τrAr, with τ1 = 2 if k = j and τ1 = 0 otherwise,and taking into account that β1i 6= 0 we may conclude that

AjPi = 2βjiA1 + · · ·

=2βji

β1i(β1iA1 + · · ·)

=2βji

β1iPi

= λijPi (32)

with λij = 2βji

β1i, for j = 2, · · · ,m− 1. Furthermore,

AmPi =βmi

β1i(β1iIn + · · ·)

= λim(β1iIn + · · ·) (33)

with λim = βmi

β1i. Concluding,

– If j ∈ {2, · · · ,m− 1} then Aj =∑m

i=1 λijPi =∑m

i=1 2βji

β1iPi.

– Am =∑m

i=1 λimPi =∑m

i=1βmi

β1iPi.

– A1 = In =∑m

i=1 Pi.

Therefore, the above computations leads to the character table of Vn rep-resented by the table 2.

• Suppose now that n is odd. Then we have the equations

A2 = C1 + Cn−1,

A3 = C2 + Cn−2,

... =...

Am = Cm−1 + Cm,

9

A1 A2 A3 · · · Am−1 Am

P1 1 2β21β11

2β31β11

· · · 2β(m−1) 1

β11

βm 1β11

P2 1 2β22β12

2β32β12

· · · 2β(m−1) 2

β12

βm2β12

......

......

......

...Pm 1 2β2m

β1m2β3m

β1m· · · 2β(m−1) m

β1m

βmm

β1m

Table 2: The character table of Vn when n is even.

and henceA2A1 = A2,A2A2 = A3 + 2In,A2Ai = Ai+1 + Ai−1, i = 3, ..,m− 1,

A2Am = Am−1 + Am.

(34)

For i ∈ {1, · · · ,m} suppose that

Pi =m∑

k=1

βkiAk (35)

and also that

A2 =m∑

k=1

λk2Pk. (36)

Multiplying both members of (35) by A2, we obtain A2Pi =∑m

k=1 βkiA2Ak,and multiplying both members of (36) by Pi, it follows that A2Pi = λi2Pi.So, we may write

m∑k=1

λi2βkiAk =m∑

k=1

βkiA2Ak. (37)

Representing A2Ak on the basis B and replacing A2Ak, for k = 1, · · · ,m,by the left hand sides of (3) and (37) we get the equalities: 2β2i = λi2β1i,

β(j−1)i + β(j+1)i = λi2βji, for j = 2, · · · ,m− 1,β(m−1)i + βmi = λi2βmi.

(38)

Consequently the determination of the coordinates of projectors Pi on thebasis B may now be done in a similar way as when n is even. Nevertheless,we will proceed with the main steps. Considering the same notations asabove we may write

Pi = β1i

m∑k=1

βkiAk, (39)

where βki = βki

β1i, for k = 1, · · · ,m. Once again, it is clear that β1i = 1

and also that all the remaining scalars βki can be obtained, recursively,

10

from (38). Representing (Pi)2 on the basis B, we have P 2i = β2

1i((β21i +∑m−1

k=2 2β2ki + 2β2

mi)In + · · ·) and, from the equality (Pi)2 = Pi, it followsthat β1iβ1i = β2

1i(β21i +

∑m−1k=2 2β2

ki + 2β2mi]. Therefore, we conclude that

β1i =β1i

β21i +

∑m−1k=2 2β2

ki + 2β2mi

. (40)

In order to determine the character table of Vn, note that the computationsAjPi, for j = 2, · · · ,m − 1, are similar to the ones when n is even, andthen λij = 2βji

β1i. For j = m we get the equality 3

AmPi =2βmi

β1i(β1iIn + · · ·)

which implies λim = 2βmi

β1i. Therefore, the Ajs representation on the basis

B′ may be written as follows.

– If j ∈ {2, · · · ,m− 1} then Aj =∑m

i=1 λijPi =∑m

i=1 2βji

β1iPi.

– Am =∑m

i=1 λimPi =∑m

i=1 2βmi

β1iPi.

– A1 = In =∑m

i=1 Pi.

Finally we get the following character table of Vn (note that, the tables 2and 3 differs only on the column corresponding to Am).

A1 A2 A3 · · · Am−1 Am

P1 1 2β21β11

2β31β11

· · · 2β(m−1) 1

β112βm 1

β11

P2 1 2β22β12

2β32β12

· · · 2β(m−1) 2

β122βm2

β12...

......

......

......

Pm 1 2β2m

β1m2β3m

β1m· · · 2β(m−1) m

β1m2βmm

β1m

Table 3: The character table of Vn when n is odd.

4 Strongly regular graphs in Vn and their spec-tra

A graph G, is a pair of sets (V (G), E(G)), where V (G) denotes the nonempty setof vertices and E(G) the set of edges. An element of E(G), which the endpointsare the vertices i and j, is denoted by ij and, in such case, we say that the vertex iis adjacent to the vertex j. If v ∈ V (G), then we call neighborhood of v the vertexset denoted by NG(v) = {w : vw ∈ E(G)}. The complement of G, denoted by G,is such that V (G) = V (G) and E(G) = {ij : i, j ∈ V (G)∧ ij 6∈ E(G)}. A graphof order n in which all pairs of vertices are adjacent is called a complete graphand denoted by Kn. On the other hand, when there are no pair of adjacent

3Note that, as before, β1i 6= 0 and, from theorem 2.1, ∀k ∈ {1, · · ·n} AmAk =∑m

r=1 τrAr,with τ1 = 2 if k = m and τ1 = 0 otherwise.

11

vertices the graph is called a null graph and denoted by Nn. In this paper weassume that G is of order n > 1, i.e., |V (G)| = n > 1 and we deal only withsimple graphs, that is, graphs with nor loops (edges with both ends in the samevertex) neither parallel edges (more than one edge between the same pair ofvertices). Consider the set of vertices of the graph G, X = {x1, x2, . . . , xk},and suppose that xixi+1 ∈ E(G) ∀i ∈ {1, . . . , k − 1}. If the vertices of X areall distinct then we say that X defines a path (and if they are all distinct butx1 and xk, which are equal, then we say that X defines a cycle). If there is noi ∈ {1, . . . , k − 2} and j > 1 such that xixi+j ∈ E(G), then X is denoted byPk (Ck) and we say that it is a chordless path (cycle). The adjacency matrix ofthe graph G is the matrix AG = (aij)n×n such that

aij ={

1 , if ij ∈ E(G)0 , otherwise.

The number of neighbors of v ∈ V (G) will be denoted by dG(v) and called degreeof v. If G is such that ∀v ∈ V (G) dG(v) = k then we say that G is k-regular.A graph G is called strongly regular if it is regular, not complete, not null and,given any two distinct vertices i, j ∈ V (G), the number of vertices which areneighbors of both i and j depends on whether i and j are adjacent or not.When G is a strongly regular graph of order n which is p-regular and any pairof adjacent vertices have a common neighbors and any two distinct non-adjacentvertices have c common neighbors, then we say that G is a (n, p, a, c)-stronglyregular graph. The chordless cycle on five vertices C5 is an example of a stronglyregular graph which is (5,2;0,1)-strongly regular.It is well known (see for instance [6]) that a graph G which is not complete andnot null is (n, p, a, c)-strongly regular iff

A2G = (a− c)AG + (p− c)In + cJn, (41)

where Jn denotes the all ones square matrix of order n. Therefore, since AG =Jn − In − AG we may conclude that a graph is strongly regular if and only ifits complement is also strongly regular. Furthermore, it is known that if G is a(n, p, a, c)-strongly regular graph then G is (n, n−p−1, n−2−2p+c, n−2p+a)-strongly regular graph.Now, assuming that n is even, we prove that the graph Γ1 such that AΓ1 =∑m−1

j=2 Aj is a strongly regular graph and then our goal of finding a stronglyregular graph (the one with adjacency matrix AΓ1) in an Euclidean Jordanalgebra (the one spanned by the matrices A1, · · · , Am) is fulfilled.

Theorem 4.1 Let Vn be the Euclidean Jordan algebra spanned by F , with neven. Then the graph Γ1 with adjacency matrix AΓ1 =

∑m−1j=2 Aj is a (n, n −

2, n− 4, n− 2)-strongly regular graph.

Proof.By construction of the matrices A1, · · · , Am we know that

m∑j=1

Aj = Jn ⇔m−1∑j=2

Aj = Jn −Am − In.

Additionally, since Am is the adjacency matrix of a 1-regular graph it follows thatJnAm = AmJn = Jn. On the other hand, theorem 2.1 implies that A2

m = In. Then

12

we may conclude that

A2Γ1 = (

m−1∑i=2

Ai)2,

= Jn(Jn −Am − In)−Am(Jn −Am − In)− In(Jn −Am − In),

= nJn − Jn − Jn − Jn + (Am)2 + Am − Jn + Am + In,

= (n− 4)Jn + (Am)2 + 2Am + In,

= (n− 4)Jn + In + 2(Jn − In −m−1∑i=2

Ai) + In,

= (n− 2)Jn − 2

m−1∑i=2

Ai,

= (n− 2)Jn − 2AΓ1 .

Then according to (41) we may conclude that Γ1 is (n, n− 2, n− 4, n− 2)-strongly regular.Since Γ1 is strongly regular, it follows that A2

Γ1is also a linear combination

of In, AΓ1(which is such that AΓ1

= Jn − In − AΓ1) and AΓ1 . Therefore,AΓ1AΓ1

= AΓ1AΓ1 and, denoting the space spanned by In, AΓ1 and AΓ1

byV ′n, we may conclude that V ′n is closed for the products AΓ1AΓ1

, A2Γ1

and A2Γ1

and this implies that V ′n is an Euclidean Jordan subalgebra of Vn.Given a (n, p, a, c)-strongly regular graph, it is well known that its strict eigeval-ues (that is the ones distinct from p) are

λ1 =a− c +

√∆

2,

λ2 =a− c−

√∆

2,

with ∆ = (a − c)2 + 4(p − c) (see [6]). Therefore, from theorem 4.1, it followsthat the spectra of AΓ1 is

σ(AΓ1) = {−2, 0, n− 2}.

A strongly regular graph is called primitive if both G and G are connected,otherwise it is called imprimitive. It is know (see [6]) that a (n, p, a, c)-stronglyregular graph is imprimitive if and only if c = 0 or c = p. Therefore the graphΓ1 of theorem 4.1 is imprimitive and from this theorem one may conclude that∀k ∈ N there is an imprimitive strongly regular graph Γ1 of order n = 2k, withparameters (2k, 2(k − 1), 2(k − 2), 2(k − 1)). Then we may conclude that Γ1 isa (2k, 1, 0, 0)-strongly regular.The spectra of Γ1 can be also obtained summing the columns of the charactertable 2, which correspond to A2, · · · , Am−1.

From the results of section 3 follows that

P1 =1n

(A1 + A2 + · · ·+ Am)

13

and then the coordinates of P1 on the basis B =< A1, A2, · · · , Am > are givenby βi1 = 1

n , i = 1, · · · ,m. Similarly, it comes that

Pm =1n

(A1 −A2 + A3 −A4 + .... + Am−1 −Am)

and βim = 1n (−1)i+1, i = 1, · · · ,m. Therefore the character table of Vn takes

the form represented on table 4.

A1 A2 A3 · · · Am−2 Am−1 Am

P1 1 2 2 · · · 2 2 1P2 1 2β22

β122β32

β12· · · 2β(m−2) 2

β122β(m−1) 2

β12

βm2β12

......

......

......

......

Pm 1 −2 2 · · · −2 2 −1

Table 4: The character table of Vn when n is even.

Now we construct other strongly regular graphs included in the EuclideanJordan algebra Vn, when n = 6k for k ∈ N. Since m = bn

2 c + 1 it comesthat m = 3k + 1. From theorem 2.1, (A2k+1)2 = An−2(2k+1−1)+1 + 2In =A6k−4k+1 + 2In = A2k+1 + 2In and then, A2

2k+1 is a linear combination of In,A2k+1 and Jn. Thus A2k+1 is the adjacency matrix of a strongly regular graphΓ2. As above, the corresponding spectra may be founded in the character tableof Vn. The table 5 presents the spectra of the strongly regular graphs Γ2 andΓ2. ∑

i∈{2,···,m}\{2k+1} Ai A2k+1

P1 2(m− 3) + 1 2P2

∑j∈{2,···,m}\{2k+1} 2βj2

β12

β(2k+1)2

β12...

......

Pm −2m2 + 2(m

2 − 1)− 1 2

Table 5: Spectra of the strongly regular Graphs Γ2 and Γ2.

Finally, another strongly regular graph Γ3, included in Vn, when n remainsequal to 6k, but with k ∈ N \ {1}, is determined setting AΓ3 = Ak+1 + A2k+1 +A3k+1. Indeed, since

(Ak+1)2 = A2k+1 + 2In,

(A2k+1)2 = A2k+1 + 2In,

(A3k+1)2 = In,

Ak+1A2k+1 = 2A3k+1 + Ak+1,

Ak+1A3k+1 = A3k+1−(K+1)+1 = A2k+1,

A2k+1A3k+1 = A3k+1−(2k+1)+1 = Ak+1,

14

it follows that (Ak+1 + A2k+1 + A3k+1)2 = 4(A2k+1 + A3k+1 + Ak+1) + 5In.Then Γ3 is (n, p, a, c)-strongly regular graph such that a− c = 4

p− c = 5c = 0

and then p = 5, a = 4 and c = 0. Therefore Γ3 is disconnected graph whoseconnected components are K6 and its complement Γ3 is a (6k, 6(k − 1), 6(k −2), 6(k − 1))-strongly regular graph with k ≥ 2. The eigenvalues of AΓ3 can beobtained by adding the columns corresponding to Ak+1, A2k+1 and A3k+1 oftable 4.

5 A Numerical Example

In these section let us study the Euclidean Jordan algebra V6 which impliesm = 4 and then that B =< A1, A2, A3, A4 >, with A1 = I6 and

A2 =

0 1 0 0 0 11 0 1 0 0 00 1 0 1 0 00 0 1 0 1 00 0 0 1 0 11 0 0 0 1 0

, A3 =

0 0 1 0 1 00 0 0 1 0 11 0 0 0 1 00 1 0 0 0 11 0 1 0 0 00 1 0 1 0 0

, A4 =

0 0 0 1 0 00 0 0 0 1 00 0 0 0 0 11 0 0 0 0 00 1 0 0 0 00 0 1 0 0 0

.

In order to determine the coordinates of the projectors P1, P2, P3 and P4

on B (Pi =∑4

k=1 βkiAk for i = 1, · · · , 4) let us consider the linear systems ofequations:

2β2i = λi2β1i,

β1i + β3i = λi2β2i,β2i + β4i = λi2β3i,

2β3i = λi2β4i,

where λ(i+1) 2 = 2 cos( (2π)i6 ), for i = 0, · · · , 3. So, according to (29), we obtain

the equalities

P1 = β11(β11A1 + β21A2 + β31A3 + β41A4) = β11(A1 + A2 + A3 + A4),

P2 = β12(β12A1 + β22A2 + β32A3 + β42A4) = β12(A1 +12A2 −

12A3 −A4),

P3 = β13(β13A1 + β23A2 + β33A3 + β43A4) = β13(A1 −12A2 −

12A3 + A4),

P4 = β14(β14A1 + β24A2 + β34A3 + β44A4) = β14(A1 −A2 + A3 −A4).

Applying (32) and (33) we get the equalities λij = 2βji

β1i, j = 2, · · · , 3 and

λi4 = β4i

β1ifrom which the character table of V4 follows.

15

A1 A2 A3 A4

P1 1 2 2 1P2 1 1 −1 −1P3 1 −1 −1 1P4 1 −2 2 −1

Table 6: The character table of V6.

Now the graphs Γ1 and Γ2 are defined by the adjacency matrices AΓ1 =A2 + A3 and AΓ2 = A3 (note that Γ3 = K6), that is,

AΓ1 =

0 1 1 0 1 11 0 1 1 0 11 1 0 1 1 00 1 1 0 1 11 0 1 1 0 11 1 0 1 1 0

and AΓ2 =

0 0 1 0 1 00 0 0 1 0 11 0 0 0 1 00 1 0 0 0 11 0 1 0 0 00 1 0 1 0 0

.

The table 7 presents the spectra of the graphs Γ1 and Γ2, both depicted infigure 1.

AΓ1 AΓ2

P1 4 2P2 0 −1P3 −2 −1P4 0 2

Table 7: Spectra of the graphs Γ1 and Γ2.

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t tt t2 5

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3 4�

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��

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tt

tt

tt

1

5

3

4

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Figure 1: Graphs Γ1 and Γ2 of V6.

References

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[3] Faraut, J. and A. Koranyi. Analysis on Symmetric Cones. Oxford Mathe-matical Monographs, Clarendon Press, Oxford (1994).

[4] Faradzev, I. A., M. H. Klin and M. E. Muzichuk. Cellular rings and groupsof automorphisms of graphs. In Faradzev, I. A. et al (eds.): Investigationsin algebraic theory of combinatorial objects. Kluwer Acad. Publ. Dordrecht,(1994): 1–152.

[5] Fiedler, F. Enumeration of Cellular Algebras Applied to Graphs with Pre-scribed Symmetrry. Ph.D. dissertation, Technische Universitat Dresden,Dresden (1998).

[6] Godsil,C.D, Algebraic Combinatorics, Chapman & Hall, New York (1993).

[7] Klin, M., C. Rucker, G. Rucker and G. Tinhofer. Algebraic Combinatoricsin Mathematical Chemestry I. Permutation Groups and Coherent (Cellu-lar) Algebras. MATHC - Communications in Mathematics and in ComputerChemestry, 40 (1999): 1–138.

[8] Koecher, M. The Minnesota Notes on Jordan Agebras and Their Applica-tions. Lecture Notes in Mathematics, 1710, Springer, Berlin (1999).

[9] Pech, C. Incidence structures and cellular algebras. Ph. D. dissertation, Tech-nische Universitat Dresden, (1998).

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