Strongly Regular Graphs Satisfying the 4-Vertex Condition

Abstract

We survey the area of strongly regular graphs satisfying the 4-vertex condition and find several new families. We describe a switching operation on collinearity graphs of polar spaces that produces cospectral graphs. The obtained graphs satisfy the 4-vertex condition if the original graph belongs to a symplectic polar space.

Appendix: Details on Ivanov’s Graphs

In Sect. 3.3 we discussed the graphs $\Gamma ^{(m)}$ from [6] and $\Sigma ^{(m)}$ from [22]. Here we give some more detail on the latter.

For $m \ge 2$, consider $V = {{\mathbb {F}}}_{2}^{2m}$ provided with the elliptic quadratic form $q(x) = x_{1}^{2} + x_{2}^{2} + x_1x_2 + x_3x_4 +... + x_{2\,m-1}x_{2\,m}$. Identify the set of projective points (1-spaces) in V with $V^* = V\setminus \{0\}$. Let $Q = \{ x \in V^* \mid q(x) = 0 \}$ and let S be the maximal t.s. subspace given by $S = \{ x \in V^* \mid x_1 = x_2 = 0 ~\textrm{and}~ x_{2i-1}=0 ~(2 \le i \le m) \}$. Then $S^\perp = \{ x \in V^* \mid x_{2i-1}=0 ~(2 \le i \le m) \}$. The graph $\Sigma ^{(m)}$ has V as vertex set, where two distinct vertices v, w are adjacent when $v-w \in (Q \cup S^\perp )\setminus S$. Let $\textrm{T}^{(m)}$ and $\Upsilon ^{(m)}$ be the induced subgraphs on the neighbors (nonneighbors) of the vertex 0. Put $R = V^* \setminus (Q \cup S^\perp )$.

Proposition

• (i) For $m \le 4$, the graphs $\Sigma ^{(m)}$ are rank 3, and are isomorphic to the complement of $VO_{2m}^-(2)$.

• (ii) For $m \ge 5$, the automorphism group of $\textrm{T}^{(m)}$ has two vertex orbits $S^\perp \setminus S$ and $Q \setminus S$, of sizes $3 \cdot 2^{m-1}$ and $2^{2\,m-1}-2^m$, respectively. For $2 \le m \le 4$, the group is rank 3, and the graph is the complement of $NO_{2m}^-(2)$.

• (iii) For $m \ge 5$, the automorphism group of $\Upsilon ^{(m)}$ has two vertex orbits S and R of sizes $2^{m-1}-1$ and $2^{2\,m-1}-2^m$, respectively. For $3 \le m \le 4$, the group is rank 3, and the graph is the complement of $O_{2m}^-(2)$.

• (iv) The $\lambda$- and $\mu$-graphs in $\Upsilon ^{(m)}$ and the $\mu$-graphs in $\textrm{T}^{(m)}$ are all regular of valency $2^{m-2}(2^{m-2}+1)$. In particular, $\Upsilon ^{(m)}$ satisfies the 4-vertex condition.

• (v) The $\lambda$-graphs in $\textrm{T}^{(m)}$ have vertices of valencies in 0, $2^{2\,m-4}-2^m$, $2^{2\,m-4}$, $2^{2\,m-3}-2^m$. Edges not in a line contained in Q have $\lambda$-graphs with a single isolated vertex and $\lambda -1$ vertices of valency $2^{2m-4}$. For edges in a line contained in Q the $\lambda$-graphs have a single vertex with valency $2^{2m-3}-2^m$, and $2^{m-3}-1$ vertices with valency $2^{2m-4}-2^m$, and the remaining $2^{2m-3}+2^{m-3}$ vertices have valency $2^{2m-4}$. In particular, $\textrm{T}^{(m)}$ satisfies the 4-vertex condition, with $\alpha = 2^{2m-5}(2^{2m-3}+2^{m-2}-1)$ and $\beta = \frac{1}{2} \mu \mu ' = 2^{2m-4} (2^{m-2}+1)^2$.

• (vi) The local graph of $\Upsilon ^{(m)}$ at a vertex $s \in S$ is isomorphic to $\Sigma ^{(m-1)}$.

Proof

(i)–(iii) This is clear, and can also be found in [22]. (iv)–(v) (the part about $\textrm{T}^{(m)}$):

Let $(v,w) = q(v+w)-q(v)-q(w)$ be the symmetric bilinear form belonging to q. Let $X = (Q \cup S^\perp )\setminus S$. Then $\textrm{T}^{(m)}$ is the graph with vertex set X, where two vertices x, y are adjacent when the projective line $\{x,y,x+y\}$ they span is contained in X. If at least one of x, y is in $S^\perp \setminus S$, then this is equivalent to $(x,y)=1$. If both are in $Q\setminus S$, then this is equivalent to ($(x,y)=0$ and $x+y \notin S$) or ($(x,y)=1$ and $x+y \in S^\perp \setminus S$).

Let x, y, z be pairwise adjacent vertices. The valency c of z in the $\lambda$-graph $\lambda (x,y)$ is the number of common neighbors of x, y, z. Distinguish several cases.

If $z = x+y$, then if $x,y,z \in Q$ we find $c = | \{x,y\}^\perp \cap (Q \setminus S) | - 3 = 2^{2\,m-3}-2^m$. If $z = x+y$ and at least one of x, y, z lies in $S^\perp$, then $c = 0$.

Now let $z \ne x+y$. The claims are true for $m \le 4$. Let $m \ge 5$ and use induction on m. Choose coordinates so that x, y, z have final coordinates 00 and let $x',y',z'$ be these points without the final two coordinates. If they have $c'$ common neighbors $w'$ in $\textrm{T}^{(m-1)}$, then we find $2c'$ common neighbors $w = (w',0,*)$. Moreover (since x, y, z are linearly independent), we find $2^{2m-5}$ common neighbors $(w',1,q'(w'))$ in Q, where $w'$ runs through all vectors with the desired inner products with $x',y',z'$. Altogether $c = 2c'+2^{2m-5}$, as claimed.

For the $\mu$-graphs the argument is similar and simpler: by the definition of adjacency three dependent vertices are pairwise adjacent, so that the case $z = x+y$ does not occur here.

(iv) (the part about $\Upsilon ^{(m)}$): Let $Y = V^*\setminus X$. Then $\Upsilon ^{(m)}$ is the graph with vertex set Y, where two vertices x, y are adjacent when the projective line $\{x,y,x+y\}$ they span is not contained in Y. The same argument as before yields the valencies of the $\lambda$- and $\mu$-graphs.

(vi) Consider the graph $\Sigma ^{(m)}$. The nonneighbors z of 0 that are neighbors of s are the vertices of the form $z=s+b$ with $z \in S \cup R$ and $b \in (Q \cup S^\perp ) \setminus S$. It follows that $s+z \in Q \setminus s^\perp$. Let $s = (0\ldots 01)$, then $Q\setminus s^\perp$ can be identified with $W = {{\mathbb {F}}}_{2}^{2\,m-2}$ via $w \rightarrow i(w)=(w,1,{\bar{q}}(w))$ for $w \in {{\mathbb {F}}}_{2}^{2\,m-2}$ and ${\bar{q}}(w)$ determined by $q(i(w))=0$. The local graph of $\Upsilon$ at s can be identified with the graph with vertices w, where $w,w'$ are adjacent when the line joining $i(w),i(w')$ has third point $(w+w',0,*) \in (Q \cup S^\perp )\setminus S$, that is, the line joining $w,w'$ has third point $w''=w+w'$ satisfying $w'' \notin T$ and $({\bar{q}}(w'')=0$ or $w'' \in T^\perp )$ where $T = \{ w \in W \mid w_1=w_2=w_3=w_5=...=w_{2\,m-3}=0 \}$. But this is $\Sigma ^{(m-1)}$. $\square$