# On small strongly regular graphs with an automorphism of order $3$

**Minisymposium:**
ASSOCIATION SCHEMES

**Content:**
A graph is a strongly regular graph with parameters $(n,k,\lambda,\mu)$
if it is a $k$-regular graph of order $n$ such that each pair of adjacent
vertices has in $G$ exactly $\lambda$ common neighbors and each pair of
non-adjacent vertices has in $G$ exactly $\mu$ common neighbors.
There exist several necessary conditions for parameter sets
$(n,k,\lambda,\mu)$ to admit an existence of a strongly regular graph.
Parameter sets which satisfy all such conditions are called feasible.
Although the existence of a strongly regular graph is solved for all
feasible parameter sets with $n\leq 50$, the complete classification is
still open for parameter sets $(37,18,8,9)$, $(41,20,9,10)$,
$(45,22,10,11)$, $(49,24,11,12)$, $(49, 18, 7,6)$ and $(50,21,8,9)$.
By exhaustive computer search we obtain the full list of strongly regular
graphs on up to $50$ vertices with an automorphism of order $3$. Further,
for parameter sets $(37,18,8,9)$, $(41,20,9,10)$, $(45,22,10,11)$,
$(49,24,11,12)$, $(49, 18, 7,6)$ and $(50,21,8,9)$ we determine a class of
strongly regular graphs which contains all the $Z_3$ graphs and is
closed under particular instances of partial switching defined by
Godsil-McKay and Behbahani-Lam-Ostergaard, respectively. As a corollary,
we significantly extend list of known regular two-graphs on at most $50$
vertices.
(Recall that a two-graph is a set of (unordered) triples chosen from a
finite vertex set X, such that every (unordered) quadruple from X contains
an even number of triples of the two-graph. A regular two-graph has the
property
that every pair of vertices lies in the same number of triples of the
two-graph.)