Generalized inequalities associated to the regularity of a strongly regular graph

Abstract

Let $G$ be a primitive strongly regular graph of order $n$ . We first define the Generalized Krein parameters of $G$ . Next we establish some inequalities on the Generalized Krein parameters of a strongly regular graph and some admissibility conditions over their Krein parameters.

Keywords

Graph theory associative algebra matrix theory

1. Introduction

Euclidean Jordan algebras had many applications on various branches of Mathematics, namely on the formalism of quantum mechanics [1], on combinatorics [2, 3, 4, 5, 6, 7], on the developing theory for interior-point methods [8, 9, 10], and on developing applications to statistics [11].

In this work we establish inequalities on the generalized Krein parameters of a strongly regular graph in the context of Euclidean Jordan algebras.

This paper is organized as follows. In the first section, we review the principal results on Euclidean Jordan algebras. In the next section we present the principal concepts on strongly regular graphs necessary for a clear exposition of this paper. In the third section, we describe the generalized Krein parameters of a strongly regular graph, after associating a three dimensional real Euclidean Jordan algebra ${\cal A}$ to it’s adjacency matrix. Finally, in the last section, we establish some inequalities on these generalized Krein parameters, like those presented on inequality Eq. (7) of Theorem 3 and we present some inequalities for some of the Krein parameters of a strongly regular graph, and present an example of a family of strongly regular graphs that permits us to conclude that the upper bound for the Krein parameter $q^{1}_{12}$ of a strongly regular graph established on Eq. (9) is satisfied. Finally we establish some Generalized Krein conditions associated to the regularity of a strongly regular graph like those established on inequalities Eq. (10) and on Eq. (15) of Theorems 4 and 5 respectively.

2. Some notions on Euclidean Jordan algebras

Good introductions to Euclidean Jordan algebras can be found in the monograph by Faraut and Korányi[12], and in Koecher’s lecture notes [13]. Now, we will present only the principal results needed for a clear understanding of this paper. Throughout this work we assume that a $n$ -dimensional real vector space $\mathcal{A}$ with a bilinear map $(x,y)\mapsto x\bullet y$ in $\mathcal{A}\times\mathcal{A}\rightarrow\mathcal{A}$ is a real Jordan algebra if $x\bullet y=y\bullet x$ and $x\bullet(x^{2\bullet}\bullet y)=x^{2\bullet}\bullet(x\bullet y)$ , where $x^{2\bullet}=x\bullet x$ . We will suppose, from now on that if $\mathcal{A}$ is a real Jordan algebra, then $\mathcal{A}$ is finite dimensional and has a unit element denoted by $\mathbf{e}$ . Let $\mathcal{A}$ be a n-dimensional real Jordan algebra. Then $\mathcal{A}$ is power associative, that is an algebra such that for any $x$ in $\mathcal{A}$ the algebra spanned by $x$ and $\mathbf{e}$ is associative. Since $\mathcal{A}$ is an associative algebra so one defines $x^{k\bullet}$ in the natural way $x^{0\bullet}=\mathbf{e},x^{1\bullet}=x$ and for any natural number $l$ one defines $x^{l\bullet}=x\bullet x^{l-1\bullet}$ .

The rank of $x$ in $\mathcal{A}$ is the least natural number $k$ such that $\{x^{0\bullet},x^{1\bullet},\ldots,x^{k\bullet}\}$ is linearly dependent and we write $\text{rank}(x)=k$ . Since $\text{rank}(x)\leqslant n$ the rank of ${\cal A}$ is defined as being the natural number $\text{rank}({\cal A})=\text{max}\{\text{rank}(x):x\in{\cal A}\}$ . An element $x$ in ${\mathcal{A}}$ is regular if $r=\text{rank}(x)=\text{rank}({\cal A})$ . Let $x$ be a regular element of ${\mathcal{A}}$ and $r=\text{rank}(x)$ . Then, there exist real scalars $a_{1}(x),a_{2}(x),\ldots,a_{r-1}(x)$ and $a_{r}(x)$ such that

$\displaystyle x^{r\bullet}-a_{1}(x)x^{r-1\bullet}+\ldots+(-1)^{r}a_{r}(x)x^{0% \bullet}=0,$ (1)

where $0$ is the zero vector of ${\mathcal{A}}$ . Taking into account (1) we conclude that the polynomial

$\displaystyle p(x,\lambda)=\lambda^{r}-a_{1}(x)\lambda^{r-1}+\ldots+(-1)^{r}a_% {r}(x)$ (2)

is the minimal polynomial of $x$ . When $x$ is not regular the minimal polynomial of $x$ has a degree less than $r$ . The roots of the minimal polynomial of $x$ are the eigenvalues of $x$ .

Let $\mathcal{A}$ be a finite dimensional associative real algebra with the bilinear map $(x,y)\mapsto x\bullet y$ . We introduce on $\mathcal{A}$ a structure of Jordan algebra by considering a new product $\star$ defined by $x\star y=(x\bullet y+y\bullet x)/2$ for all $x$ and $y$ in $\mathcal{A}$ . The product $\star$ is called the Jordan product.

A real Euclidean Jordan algebra ${\mathcal{A}}$ is a Jordan algebra with an inner product $<\cdot,\cdot>$ such that $<x\bullet y,z>=<y,x\bullet z>$ , for all $x, y$ and $z$ in ${\mathcal{A}}$ . Note that the real vector space of real symmetric matrices of order $n,\text{Sym}(n,\mathbb{R}),$ is a real Euclidean Jordan algebra endowed with the Jordan product and the inner product defined by $<x,y>=\text{trace}(xy)$ , where trace denotes the usual trace of matrices. The unit element in this case is the identity matrix of order $n$ , $I_{n}$ .

Let ${\mathcal{A}}$ be a real Euclidean Jordan algebra with unit element e. An element $c$ in ${\mathcal{A}}$ is an idempotent if $c^{2\bullet}=c$ . Two idempotents $c$ and $d$ are orthogonal if $c\bullet d=$ 0. The set $\{c_{1},c_{2},\ldots,c_{l}\}$ of non zero idempotents is a complete system of orthogonal idempotents if the following three conditions hold: $(i)$ $c^{2}_{i}=c_{i}$ , for $i=1,\ldots,l$ , $(ii)$ $c_{i}\bullet c_{j}=0$ , if $i\not=j$ , and $(iii)$ $\sum^{l}_{i=1}c_{i}=\mathbf{e}$ . An idempotent $c$ is primitive if it is a nonzero idempotent of ${\mathcal{A}}$ and if it can’t be written as a sum of two non-zero idempotents. We say that $\{c_{1},c_{2},\ldots,c_{k}\}$ is a Jordan frame if $\{c_{1},c_{2},\ldots,c_{k}\}$ is a complete system of orthogonal idempotents such that each idempotent is primitive.

.

Let $C$ be a matrix of the Euclidean Jordan algebra ${\cal A}=\text{Sym}(n,\mathbb{R})$ . If $C$ has $k$ distinct eigenvalues $\lambda_{1},\lambda_{2},\ldots,\lambda_{k-1}$ and $\lambda_{k}$ then the set $S=\{c_{1},c_{2},\ldots,c_{k}\}$ is a complete system of orthogonal idempotents of ${\cal A}$ where each idempotent $c_{i}$ for $i=1,\ldots,k$ is the projector on the eigenvector space of $C$ associated to the eigenvalue $\lambda_{i}$ defined by the equality

$\displaystyle c_{i}=\prod^{k}_{j=1,j\neq i}\frac{\lambda_{j}I_{n}-C}{\lambda_{% j}-\lambda_{i}}$

for $i=1,\ldots,k$ . We must say that they are unique, and we have $c_{1}+c_{2}+\ldots+c_{k}=I_{n},$ $\>c_{i}\bullet c_{j}=0$ for $i\neq j$ and $i,j\in\{1,\ldots,k\}$ where $\bullet$ is the Jordan product of matrices, $c^{2}_{i}=c_{i}$ for $i=1,\ldots,n$ and we have $C=\sum^{k}_{j=1}\lambda_{j}c_{j}$ .

.

Consider the Euclidean Jordan Algebra ${\cal A}=\text{Sym}(4,\mathbb{R})$ with the Jordan product $x\bullet y=\frac{xy+yx}{2}$ and the inner product $<x,y>=\text{trace}{(x\bullet y)}$ . Then

$\displaystyle S=\{a,b\}=\left\{\left[\begin{array}[]{llll}1&0&0&0\\ 0&1&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array}\right],\left[\begin{array}[]{llll}0&0&0&0\\ 0&0&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{array}\right]\right\}$

is a complete system of orthogonal idempotents of ${\cal A}$ .

First, we must say that for any element $x\in{\cal A}$ we have

$\displaystyle x\bullet x=\frac{xx+xx}{2}=\frac{2xx}{2}=x^{2}.$

So the natural powers of an element $x$ of ${\cal A}$ relatively to the product $\bullet$ coincides with the natural powers of $x$ relatively to the usual product of matrices. We must also say, that $x|y=\text{trace}(x\bullet y)=\text{trace}(xy)$ where $x y$ represents the usual product of matrices. So, since $a^{2}=a$ and $b^{2}=b$ and

$\displaystyle a\bullet b=\frac{ab+ba}{2}=\frac{\left[\begin{array}[]{llll}1&0&% 0&0\\ 0&1&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array}\right]\left[\begin{array}[]{llll}0&0&0&0\\ 0&0&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{array}\right]+\left[\begin{array}[]{llll}0&0&0&0\\ 0&0&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{array}\right]\left[\begin{array}[]{llll}1&0&0&0\\ 0&1&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array}\right]}{2}=\left[\begin{array}[]{llll}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ \end{array}\right]$

and $a+b=I_{4}$ then $S=\{a,b\}$ is a complete system of orthogonal idempotents of ${\cal A}$ . Finally, we must say, that $S_{1}=\{E_{11},E_{22},\ldots,E_{nn}\}$ is a Jordan Frame of the Euclidean Jordan algebra ${\cal A}=\text{Sym}(n,\mathbb{R})$ where for each $i$ such that $1\leqslant i\leqslant n,$ $(E_{ii})_{ii}=1$ and all the others entries of the matrix $E_{ii}$ are all null.

.

[12, p. 43]. Let ${\mathcal{V}}$ be a real Euclidean Jordan algebra. Then for $x$ in ${\mathcal{V}}$ there exist unique real numbers $\lambda_{1},\lambda_{2},\ldots,\lambda_{k},$ all distinct, and a unique complete system of orthogonal idempotents $\{c_{1},c_{2},\ldots,c_{k}\}$ such that

$\displaystyle x=\lambda_{1}c_{1}+\lambda_{2}c_{2}+\ldots+\lambda_{k}c_{k}.$ (3)

The numbers $\lambda_{j}$ ’s of Eq. (3) are the eigenvalues of $x$ and the decomposition Eq. (3) is the first spectral decomposition of $x$ .

.

[12, p. 44]. Let ${\mathcal{V}}$ be a real Euclidean Jordan algebra with $\text{rank}({\mathcal{V}})=r$ . Then for each $x$ in ${\mathcal{V}}$ there exists a Jordan frame $\{c_{1},c_{2},\ldots,c_{r}\}$ and real numbers $\lambda_{1},\ldots,\lambda_{r-1}$ and $\lambda_{r}$ such that

$\displaystyle x=\lambda_{1}c_{1}+\lambda_{2}c_{2}+\ldots+\lambda_{r}c_{r}.$ (4)

The decomposition Eq. (4) is called the second spectral decomposition of $x$ .

3. Results on strongly regular graphs

Along this paper we consider only non-empty, simple and non complete graphs. By simple graphs we mean graphs without loops and parallel edges. Strongly regular graphs were firstly introduced by Bose[14]. Let $G$ be a graph of order $n$ . We denote its vertices set by $V(G)$ and its edge set by $E(G)$ . An edge whose endpoints are $u$ and $v$ is denoted by $u v$ and, in such case, the vertices $u$ and $v$ are adjacent or neighbors. The number of vertices of $G$ , $|V(G)|$ , is called the order of $G$ . If all vertices of $G$ have $k$ neighbors, then $G$ is called a $k$ -regular graph. $G$ is called a $(n,k,\lambda,\mu)$ -strongly regular graph if is $k$ -regular and any pair of adjacent vertices have $\lambda$ common neighbors and any pair of non-adjacent vertices have $\mu$ common adjacent vertices.

Let $G$ be a $(n,k,\lambda,\mu)$ -strongly regular graph. The adjacency matrix of $G$ , $A=[a_{ij}]$ , is a binary matrix of order $n$ such that $a_{ij}=1$ , if the vertice $i$ is adjacent to $j$ and $0$ otherwise. The adjacency matrix of $G$ satisfies the equation $A^{2}=kI_{n}+\lambda A+\mu(J_{n}-A-I_{n})$ , where $J_{n}$ is the all one matrix of order $n$ .

It is well known (see, for instance, [15]) that the eigenvalues of $G$ are $k$ , $\theta$ and $\tau$ , where $\theta$ and $\tau$ are given by $\theta=(\lambda-\mu+\sqrt{(\lambda-\mu)^{2}+4(k-\mu)})/2$ and $\tau=(\lambda-\mu-\sqrt{(\lambda-\mu)^{2}+4(k-\mu)})/2$ , (see [15]).

A strongly regular graph $G$ is called a primitive if $G$ and $\overline{G}$ are connected, otherwise is called imprimitive. A $(n,k,\lambda,\mu)$ -strongly regular graph is an imprimitive strongly regular graph if and only if $\mu=0$ or $\mu=k$ . Since in this paper we only consider non complete strongly regular graphs that are primitive, we then consider only strongly regular graphs $(n,k,\lambda,\mu)$ such that $0<\mu<k$ .

4. Euclidean Jordan algebra associated to a strongly regular graph

Now, let $\text{Sym}(n,\mathbb{R}),$ be the Euclidean Jordan algebra of real Symmetric matrices of order $n$ endowed with the Jordan product $x\star y=\frac{xy+yx}{2}$ and with inner product $x|y=\text{trace}(xy),$ where $x y$ and $y x$ are the usual product of matrices $x$ and $y$ and the usual product $y$ and $x$ . Let $G$ be a $(n,k,\lambda,\mu)$ -strongly regular graph and $A$ be its adjacency matrix with the distinct eigenvalues, namely $k$ , $\theta$ and $\tau$ . Herein, $k$ and $\theta$ are the positive eigenvalues and $\tau$ is the negative eigenvalue. Now we consider the Euclidean Jordan subalgebra $\mathcal{A}$ of $\text{Sym}(n,\mathbb{R})$ , spanned by $I_{n}$ , and the natural powers of $A$ . Since $A$ has three distinct eigenvalues, then $\mathcal{A}$ is a three dimensional real Euclidean Jordan algebra with $\text{rank}(\mathcal{A})=3$ . Let $\mathcal{B}=\{E_{1},E_{2},E_{3}\}$ be the unique complete system of orthogonal idempotents of $\mathcal{A}$ associated to $A$ , with $E_{1}=1/nI_{n}+1/nA+1/n(J_{n}-A-I_{n}),$ $E_{2}=(|\tau|n+\tau-k)/(n(\theta-\tau))I_{n}+(n+\tau-k)/(n(\theta-\tau))A+(% \tau-k)/(n(\theta-\tau))(J_{n}-A-I_{n}),$ and $E_{3}=(\theta n+k-\theta)/(n(\theta-\tau))I_{n}+(-n+k-\theta)/(n(\theta-\tau))% A+(k-\theta)/(n(\theta-\tau))(J_{n}-A-I_{n}).$

Let $M_{n}(\mathbb{R})$ be the set of square matrices of order $n$ with real entries. For $B=[b_{ij}]$ , $C=[c_{ij}]$ in $M_{n}(\mathbb{R})$ , we denote by $B\bullet C=[b_{ij}c_{ij}]$ the Hadamard product of matrices $B$ and $C$ and by $B\otimes C=[b_{ij}C]$ the Kronecker product of matrices $B$ and $C$ . Now, we introduce the following compact notation for the Hadamard and the Kronecker powers of the elements of $S$ . Let $x$ , $y$ , $z$ , $\alpha$ , $\beta$ and $\gamma$ be natural numbers such that $0\leqslant\alpha,\beta,\gamma\leqslant 2$ , $x\geqslant 2$ and $\alpha<\beta$ . Then we define

$\displaystyle E^{\bullet x}_{\alpha}=\left(E_{\alpha}\right)^{\bullet x}\text{% and }E^{\otimes x}_{\alpha}=\left(E_{\alpha}\right)^{\otimes x},E^{\bullet yz% }_{\alpha\beta}=\left(E_{\alpha}\right)^{\bullet y}\bullet\left(E_{\beta}% \right)^{\bullet z}\text{ and }E^{\otimes yz}_{\alpha\beta}=\left(E_{\alpha}% \right)^{\otimes y}\otimes\left(E_{\beta}\right)^{\otimes z}.$

Since the Euclidean Jordan algebra $\mathcal{A}$ is closed under the Hadamard product and $\mathcal{B}$ is a basis of $\mathcal{A}$ , then there exist real numbers $q_{\alpha x}^{i}$ and , $q_{\alpha\beta yz}^{i}$ , such that

$\displaystyle E^{\bullet x}_{\alpha}=\sum_{i=0}^{2}q_{\alpha x}^{i}E_{i},$ (5) $\displaystyle E^{\bullet yz}_{\alpha\beta}=\sum_{i=0}^{2}q_{\alpha\beta yz}^{i% }E_{i},$ (6)

We call the parameters $q_{\alpha x}^{i}$ and $q_{\alpha\beta yz}^{i},$ involved on the equalities Eqs (5) and (6), the generalized Krein parameters of the strongly regular graph $G$ . Notice that $q_{\alpha 2}^{i}$ and $q_{\alpha\beta 11}^{i}$ are precisely the Krein parameters of $G$ already introduced. From now on, we suppose that $x$ and $y$ are natural numbers such that $x<y$ . Since the matrix $E_{0i\alpha\beta x(y-x)}$ such that

$\displaystyle E_{0i\alpha\beta x(y-x)}=E^{\otimes x}_{\alpha}\otimes E^{% \otimes(y-x)}_{\beta}+E^{\otimes(x-1)}_{\alpha}\otimes E_{\beta}\otimes E_{% \alpha}\otimes E^{\otimes(y-x-1)}_{\beta}+\ldots++E^{\otimes(y-x)}_{\beta}% \otimes E^{\otimes x}_{\alpha}+E^{\otimes y}_{\alpha}+E^{\otimes y}_{\beta}+E^% {\otimes x}_{0}\otimes E^{\otimes(y-x)}_{i}+\ldots+E^{\otimes y-x}_{i}\otimes E% ^{\otimes x}_{0}$

is an idempotent matrix and ${y\choose x}E^{\bullet x(y-x)}_{\alpha\beta}+E^{\bullet y}_{\alpha}+E^{\bullet y% }_{\beta}+{y\choose x}E^{\bullet x(y-x)}_{0i}$ is a principal submatrix of $E_{0i\alpha\beta x(y-x)}$ then we deduce the inequality Eq. (7) of Theorem 3.

.

Let $X$ be a $(n,k,\lambda,\mu)$ -strongly regular graph such that $0<\mu<k<n-1,$ with three distinct eigenvalues $k$ , $\theta$ and $\tau$ . Consider the generalized Krein parameters $q_{\alpha\beta x(y-x)}^{i},q^{i}_{\alpha y}$ and $q^{i}_{\beta y}$ as defined in Eq. (6), with $i$ , $x$ , $y$ , $\alpha$ and $\beta$ in $\mathbb{N}$ , such that $0\leqslant\alpha,\beta\leqslant 2$ , and $1\leqslant i\leqslant 2,$ $\alpha<\beta,$ $((i\neq\alpha)\wedge(i\neq\beta))\vee((0\neq\alpha)\wedge(0\neq\beta))$ and $x<y$ . Then

$\displaystyle q_{\alpha\beta x\>(y-x)}^{i}\leqslant\frac{1-q^{i}_{\alpha y}-q^% {i}_{\beta y}}{\left(\begin{array}[]{c}y\\ x\end{array}\right)}-q^{i}_{0ix(y-x)}.$ (7)

From inequality Eq. (7) of Theorem 3 we conclude that

$\displaystyle q_{\alpha\beta x\>(y-x)}^{i}\leqslant\frac{1}{\left(\begin{array% }[]{c}y\\ x\end{array}\right)}-q^{i}_{0ix(y-x)}.$ (8)

Making $x=1,y=2,\alpha=1$ and $\beta=2$ on the inequality Eq. (8) and noting that $q^{i}_{0i11}=\frac{1}{n}$ we obtain

$\displaystyle q^{1}_{1211}\leqslant\frac{1}{2}-\frac{1}{n}.$ (9)

Finally, let $G$ be a $(n,k,\lambda,\mu)$ -strongly regular graph with $(n,k,\lambda,\mu)=(2l,2l-2,2l-4,2l-2)$ for some natural number such that $l>2$ . Then, the Krein parameter of $G,\>q^{1}_{1211}$ is such that $q^{1}_{1211}=\frac{1}{2}-\frac{1}{n}$ . Therefore the Theorem 3 allowed us to introduce a good upper bound, $\frac{1}{2}-\frac{1}{n}$ , for the Krein parameter $q^{1}_{12}$ of a $(n,k,\lambda,\mu)$ -strongly regular graph.

5. Generalized Krein conditions associated to a strongly regular graph

In this section we establish Generalized Krein conditions associated to a strongly regular graph, namely we will analyse the parameters $q^{1}_{23}$ and $q^{1}_{33}$ .

.

Let $X$ be a $(n,k,\lambda,\mu)$ -strongly regular graph such that $0<\mu<k<n-1,$ with three distinct eigenvalues $k$ , $\theta$ and $\tau$ . Then, if $\frac{n}{4}<k<\frac{n}{3}$ then we conclude that

$\displaystyle 3\theta^{3}+3\theta^{2}+\theta>\frac{n}{9}.$ (10)

Proof..

Since $q^{1}_{33}=\left(\frac{\theta n+k-\theta}{n(\theta-\tau)}\right)^{3}+\left(% \frac{-n+k-\theta}{n(\theta-\tau)}\right)^{3}k+\left(\frac{k-\theta}{n(\theta-% \tau)}\right)^{3}(n-k-1)>0$ then we conclude that

$\displaystyle(\theta n+k-\theta)^{3}+(-n+k-\theta)^{3}k+(k-\theta)^{3}(n-k-1)>0.$ (11)

Now, since $k<\frac{n}{3}$ then from Eq. (11) we obtain the inequality Eq. (12).

$\displaystyle\frac{(3\theta+1)^{3}}{27}-\frac{8}{27}k+\frac{1}{27}(n-k-1)>0.$ (12)

Now, expanding the left side of Eq. (12) and after some simplification, and since $k>\frac{n}{4}$ we obtain the inequality Eq. (13).

$\displaystyle 3\theta^{3}+3\theta^{2}+\theta>\frac{n}{9}.$ (13)

∎

.

Let $X$ be a $(n,k,\lambda,\mu)$ -strongly regular graph such that $0<\mu<k<n-1,$ with three distinct eigenvalues $k$ , $\theta$ and $\tau$ . If $\theta=1$ then , if $\frac{n}{4}<k<\frac{n}{3}$ then we conclude that

$\displaystyle 63>n.$ (14)

.

Let $X$ be a $(n,k,\lambda,\mu)$ -strongly regular graph such that $0<\mu<k<n-1,$ with three distinct eigenvalues $k$ , $\theta$ and $\tau$ . Then, if $\frac{2n}{3}+1<k<\frac{3n}{4}-1$ then we conclude that

$\displaystyle\left(|\tau|-\frac{2}{3}\right)^{3}>\frac{5}{108}n.$ (15)

Proof..

Since $q^{1}_{23}\geqslant 0$ and $q^{1}_{1311}=\left(\frac{|\tau|n+\tau-k}{n(\theta-\tau)}\right)^{3}+\left(% \frac{n+\tau-k}{n(\theta-\tau)}\right)^{3}k+\left(\frac{\tau-k}{n(\theta-\tau)% }\right)^{3}(n-k-1)$ we then have the following inequality Eq. (16).

$\displaystyle(|\tau|n+\tau-k)^{3}+(n+\tau-k)^{3}k+(\tau-k)^{3}(n-k-1)\geqslant 0.$ (16)

Now, since $\frac{2n}{3}+1<k<\frac{3n}{4}-1$ and after some algebraic manipulation of Eq. (16) we deduce Eq. (17).

$\displaystyle\left(|\tau|n-\frac{2n}{3}\right)^{3}+(n-\frac{2n}{3})^{3}\frac{3% n}{4}>\left(\frac{2n}{3}\right)^{3}\frac{n}{4}.$ (17)

Then, from Eq. (17) we conclude that:

$\displaystyle\left(|\tau|-\frac{2}{3}\right)^{3}n^{3}+\frac{3}{108}n^{4}>\frac% {1}{4}\frac{8}{27}n^{4}.$ (18)

Finally, after dividing both sides of Eq. (18) by $n^{3}$ we obtain the inequality Eq. (19)

$\displaystyle\left(|\tau|-\frac{2}{3}\right)^{3}+\frac{3}{108}n>\frac{8}{108}n.$ (19)

And, finally we conclude that $(|\tau|-\frac{2}{3})^{3}>\frac{5}{108}n$ . ∎

6. Conclusions and experimental results

In this section we consider some examples of parameters sets $(n,k,\lambda,\mu)$ that verify the inequality Eq. (10) or the inequality Eq. (15). In Table 1 we present the parameters sets $P_{1}=$ (400, 114, 8, 42), $P_{2}=$ (625, 192, 20, 76), $P_{3}=$ (650, 177, 8, 63), $P_{4}=$ (650, 472, 357, 304), $P_{5}=$ (900, 290, 37, 120) and $P_{6}=$ (1024, 385, 36, 210). For each set of parameters we present the order, the regularity, the positive eigenvalue $\theta,$ the negative eigenvalue $\tau$ and the parameters $q_{\theta\>n}$ and $q_{\tau\>n}$ defined such that $q_{\theta\>n}=3\theta^{3}+3\theta^{2}+\theta-\frac{n}{9}$ and $q_{\tau\>n}=(|\tau|-\frac{2}{3})^{3}-\frac{5}{108}n$ respectively.

Table 1

Numerical results

$P_{i}/\text{Parameters}$	$n$	$k$	$\lambda$	$\mu$	$\theta$	$\tau$	$q_{\theta n}$	$q_{\tau n}$
$P_{1}$	400	114	8	42	2	$-$ 36	$-$ 6.4	$-$
$P_{2}$	625	192	20	76	2	$-$ 58	$-$ 31.4	$-$
$P_{3}$	650	177	8	63	2	$-$ 57	$-$ 34.2	$-$
$P_{4}$	650	472	357	304	56	$-$ 4	$-$	$-$ 17.4
$P_{5}$	900	290	37	120	2	$-$ 85	$-$ 62	$-$
$P_{6}$	1024	385	36	210	1	$-$ 175	$-$ 106	$-$

The data presented on Table 1 confirm the results presented on the inequalities Eqs (10) and (15) in Theorems 4 and 5 respectively.

Footnotes

Acknowledgments

Luis 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 PT2020.

References

Jordan

Neumann

J.V.

and Wigner

, On an algebraic generalization of the quantum mechanical formalism, Annals of Mathematics 35 (1934), 29–64.

Vieira

L.A.

, Binomial Hadamard series and inequalities on the parameters of a strongly regular graph, Applied Mathematics 9 (2018), 1055–1071.

Cardoso

D.M.

and Vieira

L.A.

, Euclidean Jordan algebras with strongly regular graphs, Journal of Mathematical Sciences 120 (2004), 881–894.

Mano

V.M.

and Vieira

L.A

, Admissibility conditions and asymptotic behavior of strongly regular graphs, International Journal of Mathematical Models and Methods in Applied Sciences Methods 6 (2011), 1027–1033.

Mano

V.M.

and Vieira

L.A

, Alternating Schur series and necessary conditions for the existence of strongly regular graphs, International Journal of Mathematical Models and Methods in Applied Sciences Methods 8 (2014), 256–261.

Mano

V.M.

Martins

E.A

and Vieira

L.A

, On generalized binomial series and strongly regular graphs, Proyecciones Journal of Mathematics 4 (2013), 393–408.

Vieira

L.A

and Mano

V.M.

, Genereralized Krein parameters of a strongly regular, Applied Mathematics 6 (2015), 37–45.

Cardoso

D.M.

and Vieira

L.A.

, On the Optimal parameter of a self-concordant barrier over a symmetric cone, European Journal of Operational Research 169 (2006), 1148–1157.

Faybusovich

, Linear systems in Jordan algebras and primal dual interior point algorithms, Journal of Computational and Applied Mathematics 86 (1997), 149–175.

10.

Faybusovich

, Euclidean Jordan algebras and interior point algorithms, Positivity 1 (1997), 331–357.

11.

Massam

and Neher

, Estimation and testing for lattice conditional independence models on Euclidean Jordan algebras, Annals of Statistics 1 (1998), 1051–1082.

12.

Faraut

and Korányi

, Analysis on symmetric cones, Oxford Sciences Publications, Oxford, 1994.

13.

Koecher

, The minnesota notes on Jordan algebras and their applications, Springer Verlag, Berlin, 1999.

14.

Bose

R.C.

, Strongly regular graphs, partial geometries and partially balanced designs, Pacific J. Math 13 (1963), 384–419.

15.

Godsil

and Royle

, Algebraic graph theory, Springer Verlag, New York, 2001.