Abstract
We survey the area of strongly regular graphs satisfying the 4vertex 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 4vertex condition if the original graph belongs to a symplectic polar space.
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Appendix: Details on Ivanov’s Graphs
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 \(q(x) = x_{1}^{2} + x_{2}^{2} + x_1x_2 + x_3x_4 +... + x_{2\,m1}x_{2\,m}\). Identify the set of projective points (1spaces) 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 \(S = \{ x \in V^* \mid x_1 = x_2 = 0 ~\textrm{and}~ x_{2i1}=0 ~(2 \le i \le m) \}\). Then \(S^\perp = \{ x \in V^* \mid x_{2i1}=0 ~(2 \le i \le m) \}\). The graph \(\Sigma ^{(m)}\) has V as vertex set, where two distinct vertices v, w are adjacent when \(vw \in (Q \cup S^\perp )\setminus S\). 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 \(3 \cdot 2^{m1}\) and \(2^{2\,m1}2^m\), 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 \(2^{m1}1\) and \(2^{2\,m1}2^m\), 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 \(2^{m2}(2^{m2}+1)\). In particular, \(\Upsilon ^{(m)}\) satisfies the 4vertex condition.

(v)
The \(\lambda \)graphs in \(\textrm{T}^{(m)}\) have vertices of valencies in 0, \(2^{2\,m4}2^m\), \(2^{2\,m4}\), \(2^{2\,m3}2^m\). Edges not in a line contained in Q have \(\lambda \)graphs with a single isolated vertex and \(\lambda 1\) vertices of valency \(2^{2m4}\). For edges in a line contained in Q the \(\lambda \)graphs have a single vertex with valency \(2^{2m3}2^m\), and \(2^{m3}1\) vertices with valency \(2^{2m4}2^m\), and the remaining \(2^{2m3}+2^{m3}\) vertices have valency \(2^{2m4}\). In particular, \(\textrm{T}^{(m)}\) satisfies the 4vertex condition, with \(\alpha = 2^{2m5}(2^{2m3}+2^{m2}1)\) and \(\beta = \frac{1}{2} \mu \mu ' = 2^{2m4} (2^{m2}+1)^2\).

(vi)
The local graph of \(\Upsilon ^{(m)}\) at a vertex \(s \in S\) is isomorphic to \(\Sigma ^{(m1)}\).
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 \(c =  \{x,y\}^\perp \cap (Q \setminus S)   3 = 2^{2\,m3}2^m\). 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 \(\textrm{T}^{(m1)}\), then we find \(2c'\) common neighbors \(w = (w',0,*)\). Moreover (since x, y, z are linearly independent), we find \(2^{2m5}\) common neighbors \((w',1,q'(w'))\) in Q, where \(w'\) runs through all vectors with the desired inner products with \(x',y',z'\). Altogether \(c = 2c'+2^{2m5}\), 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 \(W = {{\mathbb {F}}}_{2}^{2\,m2}\) via \(w \rightarrow i(w)=(w,1,{\bar{q}}(w))\) for \(w \in {{\mathbb {F}}}_{2}^{2\,m2}\) 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 \(T = \{ w \in W \mid w_1=w_2=w_3=w_5=...=w_{2\,m3}=0 \}\). But this is \(\Sigma ^{(m1)}\). \(\square \)
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Brouwer, A.E., Ihringer, F. & Kantor, W.M. Strongly Regular Graphs Satisfying the 4Vertex Condition. Combinatorica 43, 257–276 (2023). https://doi.org/10.1007/s0049302300005y
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DOI: https://doi.org/10.1007/s0049302300005y