New families of strongly regular graphs

A strongly regular graph srg$(v,k,\lambda,\mu)$, is a graph with $v$ vertices such that each vertex lies on $k$ edges; any two adjacent vertices have exactly $\lambda$ common neighbours; and any two non-adjacent vertices have exactly $\mu$ common neighbours. We consider the strongly regular graphs constructed from a non-singular quadric $\mathscr{Q}_{n}$ in ${\rm PG}(n,q)$. The point-graph $\Gamma_{\mathscr{Q}_{n}}$ of $\mathscr{Q}_{n}$ has vertices corresponding to the points of $\mathscr{Q}_{n}$. Two vertices in $\Gamma_{\mathscr{Q}_{n}}$ are adjacent if the corresponding points of $\mathscr{Q}_{n}$ lie on a line contained in $\mathscr{Q}_{n}$. It is well known (see for example [3]) that $\Gamma_{\mathscr{Q}_{n}}$ is a strongly regular graph. In this article we let $q=2$, and construct from $\Gamma_{\mathscr{Q}_{n}}$ approximately $n/2$ new strongly regular graphs with the same parameters as $\Gamma_{\mathscr{Q}_{n}}$ (see Table 3 for a precise count).

This article proceeds as follows. Section 2 contains several preliminary results we need. Section 3 describes our construction of a series of infinite families of strongly regular graphs, the proof of the construction is given in Section 4. In Section 5, we classify and count the maximal cliques in the new graphs. In Section 6 we prove that our construction yields new families of strongly regular graphs. Finally, in Section 7, we determine the automorphism group of the new graphs.

In previous work, Kantor [8] constructed a strongly regular graph from $\Gamma_{\mathscr{Q}_{n}}$ with the same parameters in the case when $\mathscr{Q}_{n}$ contains a spread. Kantor conjects that his graph is not isomorphic to $\Gamma_{\mathscr{Q}_{n}}$. We show in Section 6.1 that the graph constructed by Kantor is not isomorphic to any of our new graphs. Abiad and Haemers [1] construct several strongly regular graphs from the symplectic graph over ${\rm GF}(2)$. The dual of these graphs have the same parameters as the point-graph of a non-singular parabolic quadric, so $n$ is even. It is not known if these graphs are isomorphic to our examples with $n$ even.

In [5], Godsil and McKay take a graph $\Gamma$, and use a vertex partition to construct a new graph $\Gamma^{\prime}$ that has the same spectrum as $\Gamma$. It is well-known (see for example [4]) that if a graph $\Gamma^{\prime}$ has the same spectrum as a strongly regular graph $\Gamma$, then $\Gamma^{\prime}$ is also strongly regular with the same parameters as $\Gamma$. Specialising the Godsil-McKay construction to a partition of size two in a strongly regular graph gives the following result.

Let $\mathscr{Q}_{n}$ be a non-singular quadric in ${\rm PG}(n,q)$. The projective index $g$ of $\mathscr{Q}_{n}$ is the dimension of the largest subspace contained in $\mathscr{Q}_{n}$. A $g$-space contained in $\mathscr{Q}_{n}$ is called a generator of $\mathscr{Q}_{n}$. If $n=2r$ is even, then a non-singular quadric is called a parabolic quadric, denoted $\mathcal{P}_{2r}$, which has projective index $g=r-1$. If $n=2r+1$ is odd, then there are two types of non-singular quadrics: the elliptic quadric denoted ${\cal E}_{2r+1}$ has projective index $g=r-1$; and the hyperbolic quadric denoted $\mathcal{H}_{2r+1}$ has projective index $g=r$. The points and generators of $\mathscr{Q}_{n}$ also form a polar space of rank $g+1$. We repeatedly use the following two properties of quadrics and polar spaces, see [7, Chapter 22] for more information on quadrics, and [7, Section 26.1] for more information on polar spaces.

We begin with a small example to illustrate the general construction.

We now give our general construction of a series of infinite families of strongly regular graphs. This construction generalises Example 3.1. First we define a partition of the vertices of the point-graph of $\mathscr{Q}_{n}$.

Note that if $s=g$, then there are no points of type (ii), so we need $s<g$. We will show that the partition $\{\mathcal{X}_{s},\mathcal{Y}_{s}\}$, $0\leq s<g$, is a Godsil-McKay partition if and only if $q=2$. By [7, Theorem 26.6.6], the group fixing $\mathscr{Q}_{n}$ is transitive on the subspaces of dimension $s$ contained in $\mathscr{Q}_{n}$. So for each $s$, $0\leq s<g$, we can use Result 2.1 to construct a unique strongly regular graph $\Gamma_{s}$ from $\Gamma$. We state the main result here, and give the proof in Section 4.

We show in Section 6 that $\Gamma_{0}\cong\Gamma$, and that for each $n$, $\Gamma_{s}$, $0\leq s<g$ are $g-1$ non-isomorphic graphs.

Throughout this section, let $\mathscr{Q}_{n}$ be a non-singular quadric in ${\rm PG}(n,q)$ of projective index $g$, and let $\alpha_{s}$ be a subspace of dimension $s$, $0\leq s<g$, contained in $\mathscr{Q}_{n}$. Let $\Gamma$ be the point-graph of $\mathscr{Q}_{n}$, and let $\{\mathcal{X}_{s},\mathcal{Y}_{s}\}$ be the partition of the vertices of $\Gamma$ (and so of the points of $\mathscr{Q}_{n}$) defined in Definition 3.2. We will show that $\{\mathcal{X}_{s},\mathcal{Y}_{s}\}$ satisfies Conditions I and II of Result 2.1. First we count the points in $\mathcal{X}_{s}$.

Proof  We prove this in the case $\mathscr{Q}_{n}$ is ${\cal E}={\cal E}_{2r+1}$, which has projective index $g=r-1$ and point-graph denoted $\Gamma_{\cal E}$. The cases when $\mathscr{Q}_{n}$ is $\mathcal{H}_{2r+1}$ and $\mathcal{P}_{2r}$ are proved in a very similar manner.

By [7, Theorem 22.5.1], the number of subspaces of dimension $s$ contained in ${\cal E}$ is

Moreover, replacing ‘$s$’ by ‘$s+1$’ in this equation gives the number of subspaces of dimension $s+1$ contained in ${\cal E}$. By [6, Theorem 3.1], the number of subspaces of dimension $s$ in a subspace of dimension $s+1$ is $\big{(}q^{s+2}-1\big{)}/\big{(}q-1\big{)}$. By [7], the number of subspaces of dimension $s+1$ that contain $\alpha_{s}$ and are contained in ${\cal E}$ is a constant. To calculate it, we count ordered pairs $(\Pi,\Sigma)$ where $\Pi$ is an $s$-dimensional subspace contained in ${\cal E}$, $\Sigma$ is an $(s+1)$-dimensional subspace contained in ${\cal E}$, and $\Pi\subset\Sigma$. This count gives the number of subspaces of dimension $s+1$ that contain $\alpha_{s}$ and are contained in ${\cal E}$ is

Each of these subspace contains $q^{s+1}$ points that are not in $\alpha_{s}$. Hence $|\mathcal{X}_{s}|=xq^{s+1}$ as required. $\square$

We now show that $\{\mathcal{X}_{s},\mathcal{Y}_{s}\}$ satisfies Condition I of Result 2.1.

Proof  We prove this in the case $\mathscr{Q}_{n}$ is ${\cal E}={\cal E}_{2r+1}$, which has projective index $g=r-1$ and point-graph denoted $\Gamma_{\cal E}$. The cases when $\mathscr{Q}_{n}$ is $\mathcal{H}_{2r+1}$ and $\mathcal{P}_{2r}$ are proved in a very similar manner.

Let $Q$ be a vertex in $\mathcal{X}_{s}$, we need to count the number of vertices in $\mathcal{X}_{s}$ that are adjacent to $Q$. Recall that $\mathcal{X}_{s}$ consists of vertices of type (ii), so in ${\rm PG}(2r+1,q)$, $Q$ is a point of the quadric ${\cal E}$, and the $(s+1)$-dimensional space $\Sigma=\langle Q,\alpha_{s}\rangle$ is contained in ${\cal E}$. A vertex $Q^{\prime}$ in $\mathcal{X}_{s}$ is adjacent to $Q$ if the line $QQ^{\prime}$ is contained in ${\cal E}$. We partition the lines of ${\cal E}$ through $Q$ into three families: ${\cal F}_{1}$ contains the lines of ${\cal E}$ through $Q$ that lie in $\Sigma$; ${\cal F}_{2}$ contains the lines of ${\cal E}$ through $Q$ (not in ${\cal F}_{1}$) that lie in an $(s+2)$-dimensional subspace that contains $\Sigma$ and is contained in ${\cal E}$; and ${\cal F}_{3}$ contains the remaining lines of ${\cal E}$ through $Q$.

We first look at ${\cal F}_{1}$. The number of lines in ${\cal F}_{1}$ equals the number of lines through a point in an $(s+1)$-dimensional subspace, so by [6, Theorem 3.1],

Each of the lines in ${\cal F}_{1}$ contains the point $Q$ and meets $\alpha_{s}$ in one point. So each line in ${\cal F}_{1}$ gives rise to $q-1$ vertices in $\mathcal{X}_{s}$ which are adjacent to $Q$ in the graph $\Gamma^{*}$. In total, ${\cal F}_{1}$ contributes $(q-1)\times|{\cal F}_{1}|=(q^{s+1}-1)$ neighbours of $Q$ in $\Gamma^{*}$.

Next we look at ${\cal F}_{2}$. Replacing ‘$s$’ by ‘$s+1$’ in (1) gives the number of subspace of dimension $s+2$ that contain the $(s+1)$-space $\Sigma=\langle Q,\alpha_{s}\rangle$ and are contained in ${\cal E}$ is $(q^{r-s-1}+1)(q^{r-s-2}-1)/(q-1)$. Similarly, (2) can be generalised to show that the number of lines through $Q$ that lie in a subspace of dimension $s+2$, and do not lie in the $(s+1)$-space $\Sigma$ is $\Big{(}(q^{s+2}-1)/(q-1)\Big{)}-\Big{(}(q^{s+1}-1)/(q-1)\Big{)}=q^{s+1}$. Hence

Each line in ${\cal F}_{2}$ contains one point of $\Sigma$, and the remaining $q$ points correspond to $q$ vertices that lie in $\mathcal{X}_{s}$ (and are not considered in ${\cal F}_{1}$). That is, each line in ${\cal F}_{2}$ contributes $q$ neighbours to $Q$ in the graph $\Gamma^{*}$. So in total, ${\cal F}_{2}$ contributes $q\times|{\cal F}_{2}|=q^{s+2}(q^{r-s-1}+1)(q^{r-s-2}-1)/(q-1)$ neighbours to $Q$ in the graph $\Gamma^{*}$.

Finally we look at ${\cal F}_{3}$. Let $\ell$ be a line in ${\cal F}_{3}$, so $\ell$ contains $Q$, but the $(s+2)$-space $\Pi=\langle\alpha_{s},\ell\rangle$ is not contained in ${\cal E}$. Suppose that $\ell$ contains another point $Q^{\prime}$ that corresponds to a vertex in $\mathcal{X}_{s}$. Then $\Pi\cap{\cal E}$ contains the two distinct $(s+1)$-dimensional subspaces $\Sigma=\langle\alpha_{s},Q\rangle$ and $\Sigma^{\prime}=\langle\alpha_{s},Q^{\prime}\rangle$. As $\Pi$ is not contained in ${\cal E}$, $\Pi$ meets ${\cal E}$ in exactly the two $(s+1)$-spaces $\Sigma$ and $\Sigma^{\prime}$. Thus $\ell=QQ^{\prime}$ is not a line of ${\cal E}$, and so $\ell$ contains exactly two points $Q,Q^{\prime}$ that are vertices of $\mathcal{X}_{s}$, moreover they are not adjacent in $\Gamma^{*}$. Thus ${\cal F}_{3}$ contributes 0 neighbours to $Q$ in the graph $\Gamma^{*}$.

Summing the neighbours of $Q$ in $\Gamma^{*}$ obtained from the families ${\cal F}_{1},{\cal F}_{2},{\cal F}_{3}$ gives the required result. Note that if $s=g-1$, so $s=r-2$, then $|{\cal F}_{2}|=0$, and the degree of $\Gamma^{*}$ is $q^{r-1}-1$. $\square$

Now we look at Condition II of Result 2.1. Note that throughout the proofs in this article, we consistently use $P,P^{\prime}$ to denote points of type (i); $Q,Q^{\prime}$ to denote points of type (ii); and $R,R^{\prime}$ to denote points of type (iii).

Proof  We prove this in the case $\mathscr{Q}_{n}$ is ${\cal E}={\cal E}_{2r+1}$, which has projective index $g=r-1$ and point-graph denoted $\Gamma_{\cal E}$. The cases when $\mathscr{Q}_{n}$ is $\mathcal{H}_{2r+1}$ and $\mathcal{P}_{2r}$ are proved in a very similar manner.

We need to show that in the graph $\Gamma_{\cal E}$, each vertex in $\mathcal{Y}_{s}$ is adjacent to $0$, $\frac{1}{2}|\mathcal{X}_{s}|$ or $|\mathcal{X}_{s}|$ vertices in $\mathcal{X}_{s}$. There are two cases to consider since the vertices in $\mathcal{Y}_{s}$ are of type (i) or (iii). First consider a vertex $P$ in $\mathcal{Y}_{s}$ of type (i). Let $Q\in\mathcal{X}_{s}$, so $Q$ is a vertex of type (ii). Hence in ${\rm PG}(2r+1,q)$, $P\in\alpha_{s}$ and $Q$ lies in an $(s+1)$-space $\Pi$ with $\alpha_{s}\subset\Pi\subset{\cal E}$. Hence $PQ$ is a line of ${\cal E}$, and so $P$ and $Q$ are adjacent vertices in $\Gamma_{\cal E}$. That is, each vertex of type (i) in $\mathcal{Y}_{s}$ is adjacent to each of the $|\mathcal{X}_{s}|$ vertices in $\mathcal{X}_{s}$.

Now consider a vertex $R$ in $\mathcal{Y}_{s}$ of type (iii). We count the number of vertices $Q$ in $\mathcal{X}_{s}$ for which $RQ$ is a line of ${\cal E}$. We will show that this number is not 0 or $|\mathcal{X}_{s}|$, and further, is $\frac{1}{2}|\mathcal{X}_{s}|$ if and only if $q=2$. Let $\Sigma$ be a subspace of ${\cal E}$ of dimension $s+1$ that contains $\alpha_{s}$. So $\Sigma\backslash\alpha_{s}$ consists of points of type (ii), hence $R\notin\Sigma$. Consider the $(s+2)$-space $\Pi=\langle\Sigma,R\rangle$. As $\alpha_{s}\subset\Sigma$, we have $\langle\alpha_{s},R\rangle\subset\Pi$. As $R$ is of type (iii), $\langle\alpha_{s},R\rangle$ is not contained in ${\cal E}$. Hence $\Pi$ is not contained in ${\cal E}$. So $\Pi\cap{\cal E}$ contains the $(s+1)$-space $\Sigma$ and the point $R\notin\Sigma$. Hence by Result 2.2, $\Pi\cap{\cal E}$ is two distinct $(s+1)$-spaces. That is, $\Pi\cap{\cal E}=\{\Sigma,\Sigma^{\prime}\}$ where $\Sigma^{\prime}$ is an $(s+1)$-space that contains $R$. As $R$ is type (iii), $\Sigma^{\prime}$ does not contain $\alpha_{s}$. Hence $\Sigma\cap\Sigma^{\prime}$ is an $s$-space distinct from $\alpha_{s}$, and so $\Sigma\cap\Sigma^{\prime}\cap\alpha_{s}$ is a space of dimension $s-1$. Let $Q$ be a point in $\Sigma^{\prime}\cap\Sigma$, $Q\notin\alpha_{s}$, so $Q$ has type (ii). As $Q,R\in\Sigma^{\prime}\subset{\cal E}$, the line $m=QR$ is a line of ${\cal E}$.

Suppose the line $m=QR$ contains a second point $Q^{\prime}$ of type (ii). So $\langle\alpha_{s},Q^{\prime}\rangle$ is an $(s+1)$-space contained in ${\cal E}$. Thus $\Pi$ contains three distinct $(s+1)$-spaces of ${\cal E}$, namely $\Sigma,\Sigma^{\prime},\langle\alpha_{s},Q^{\prime}\rangle$, contradicting Result 2.2. Thus $m$ contains exactly one point of type (ii), namely $Q$, and the rest of the points on $m$ are type (iii). Hence in the graph $\Gamma_{\cal E}$, the line $m$ gives rise to one neighbour of $R$ that lies in $\mathcal{X}_{s}$, namely $Q$. Thus each point of $\Sigma^{\prime}\cap\Sigma$ not in $\alpha_{s}$ gives rise to exactly one vertex in $\mathcal{X}_{s}$ that is a neighbour of $R$. This is true for every $(s+1)$-space $\Sigma$ with $\alpha_{s}\subset\Sigma\subset{\cal E}$. Moreover, each neighbour of $R$ in $\mathcal{X}_{s}$ corresponds to a point of ${\cal E}$ that lies in exactly one such $(s+1)$-space, so arises exactly once in this way. Hence the number of neighbours of $R$ that lie in $\mathcal{X}_{s}$ equals the number of points of ${\cal E}\backslash\alpha_{s}$ that lie in an $(s+1)$-space $\Sigma$ with $\alpha_{s}\subset\Sigma\subset{\cal E}$. We count these points.

Firstly, the number of $(s+1)$-dimensional spaces that contain $\alpha_{s}$ and are contained in ${\cal E}$ is given in (1). Secondly, let $\Sigma$ be an $(s+1)$-space containing $\alpha_{s}$, and $\Sigma^{\prime}$ an $(s+1)$-space that meets $\Sigma$ in an $s$-space not containing $\alpha_{s}$. Then the number of points in $\Sigma\cap\Sigma^{\prime}$ which are not in $\alpha_{s}$ is $\big{(}(q^{s+1}-1)/(q-1)\big{)}-\big{(}(q^{s}-1)/(q-1)\big{)}=q^{s}$. Hence in the graph $\Gamma_{\cal E}$, there are

vertices in $\mathcal{X}_{s}$ that are neighbours of $R$. To satisfy Condition II of Result 2.1, we need $y\in\{0,\ \frac{1}{2}|\mathcal{X}_{s}|,\ |\mathcal{X}_{s}|\}$. Now $y=0$ if and only if $r-s-1=0$, which does not occur as $s<g=r-1$. Further, $|\mathcal{X}_{s}|$ is calculated in Lemma 4.1, and $y<|\mathcal{X}_{s}|$. Using Lemma 4.1, $y=|\mathcal{X}_{s}|/2$ if and only if $q=2$.

Thus the vertices in $\mathcal{Y}_{s}$ of type (i) are adjacent to $|\mathcal{X}_{s}|$ of the vertices in $\mathcal{X}_{s}$. Further, the vertices in $\mathcal{Y}_{s}$ of type (iii) are not adjacent to 0 or all the vertices of $\mathcal{X}_{s}$, and are adjacent to $\frac{1}{2}|\mathcal{X}_{s}|$ of the vertices in $\mathcal{X}_{s}$ if and only if $q=2$. That is, Condition II of Result 2.1 is satisfied in for the partition $\{\mathcal{X}_{s},\mathcal{Y}_{s}\}$ of $\Gamma_{\cal E}$ if and only if $q=2$. $\square$

It is now straightforward to prove Theorem 3.3.

Proof of Theorem 3.3 Let $\mathscr{Q}_{n}$ be a non-singular quadric of ${\rm PG}(n,2)$ with projective index $g$. Let $s$ be an integer with $0\leq s<g$, let $\alpha_{s}$ be a $s$-space contained in $\mathscr{Q}_{n}$, and let $\{\mathcal{X}_{s},\mathcal{Y}_{s}\}$ be the partition given in Definition 3.2. By Lemmas 4.2 and 4.3, the partition $\{\mathcal{X}_{s},\mathcal{Y}_{s}\}$ satisfies Conditions I and II of Result 2.1(1). Hence we can use Result 2.1(2) to construct a graph $\Gamma_{s}$. Note that as the group fixing $\mathscr{Q}_{n}$ is transitive on the $s$-spaces of $\mathscr{Q}_{n}$, $0\leq s\leq g$, different choices of the subspace $\alpha_{s}$ give rise to the same (up to isomorphism) graph. So for any $s$, $0\leq s<g$, the graph $\Gamma_{s}$ is a strongly regular graph with the same parameters as $\Gamma$. $\square$

It is useful to note that the proof of Lemma 4.3 gives a description of the edges in the graph $\Gamma_{s}$. That is, let $P,P^{\prime}$ be vertices of type (i), $Q,Q^{\prime}$ vertices of type (ii), and $R,R^{\prime}$ vertices of type (iii). Then $\{P,P^{\prime}\}$, $\{P,Q\}$, $\{P,R\}$, $\{Q,Q^{\prime}\}$, $\{R,R^{\prime}\}$ are edges of $\Gamma_{s}$ if $PP^{\prime}$, $PQ$, $PR$, $QQ^{\prime}$, $RR^{\prime}$ are lines of $\mathscr{Q}_{n}$ respectively; and $\{Q,R\}$ is an edge of $\Gamma_{s}$ if $QR$ is a 2-secant of $\mathscr{Q}_{n}$. In summary, we have:

In Section 5.1, we classify the maximal cliques in the graph $\Gamma_{s}$, and in Section 5.2, we count them.