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\markboth{
Marija Maksimovi\'c and Sanja Rukavina}{New regular two-graphs on 38 and 42 vertices}
\title[How to use the {\sf mc.cls} class file]{\bf New regular two-graphs on 38 and 42 vertices}

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%\author[D. Juki\'c]{Dragan Juki\'c\corrauth}
%\address{Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, HR-31 000 Osijek, Croatia}
%\email{{\tt jukicd@mathos.hr} (D. Juki\'c)}

\author[M.\,Maksimovi\'c, S.\,Rukavina] {Marija Maksimovi\'c \corrauth and Sanja Rukavina}
\address{Faculty of Mathematics, University of Rijeka, Radmile Matej\v ci\' c, HR-51\,000 Rijeka, Croatia}
\emails{{\tt mmaksimovic@math.uniri.hr}\,\,(M.\,Maksimovi\'c), {\tt sanjar@math.uniri.hr} (S.\,Rukavina)}
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\begin{abstract}



All regular two-graphs having up to $36$ vertices are known, and the first open case is the enumeration of two-graphs on $38$ vertices. It is known that there are at least $191$ regular two-graphs on $38$ vertices and at least $18$ regular two-graphs on $42$ vertices. The number of descendants of these two-graphs is $6760$ and $120$, respectively.

In this paper, we classify strongly regular graphs with parameters $(41,20,9,10)$ having nontrivial automorphisms and show that there are exactly $7152$ such graphs.  We enumerate all regular two-graphs on $38$ and $42$ vertices with at least one descendant whose full automorphism group is nontrivial and establish that there are at least $194$ regular two-graphs on $38$ vertices and at least $752$ regular two-graphs on $42$ vertices.
Furthermore, we construct descendants with a trivial automorphism group of the newly constructed two-graphs and increase the number of known strongly regular graphs with parameters $(37,18,8,9)$ and $(41,20,9,10)$ to $6802$ and $18439$m respectively. This significantly increases the number of known strongly regular graphs with parameters $(41,20,9,10)$.

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\keywords{regular two-graph, strongly regular graph, automorphism group, orbit matrix}

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\section{Introduction}
According to \cite{taylor}, the concept of regular two-graphs was introduced by G.\,Higman to study a doubly transitive representation of the third Conway's sporadic simple group $Co_3$. The connection between regular two-graphs and strongly regular graphs was established by Taylor in \cite{taylor}, and Bussemaker, Mathon and Seidel made the first step towards classifying regular two-graphs on at most $50$ vertices and classified all regular two-graphs on $v < 30$ vertices (see \cite{bms,taylor}).
Their results were followed by Spence and McKay in
\cite{spence36} and \cite{McKay-Spence}. To the best of our  knowledge, the number of known regular two-graphs on up to $42$ vertices is given in Table \ref{numberup42}, where $n$ and $N(n)$ denote the number of vertices and the number of known regular two-graphs, respectively, and the bar on the number indicates that the classification is completed.

\begin{center}
 \begin{table}[h]%[htpb!]
 \[ { \begin{tabular}{c|ccccccccccc}
  \hline
	$n$  &  6 & 10& 14&16& 18& 26&28&30&36&38&42
\cr
$N(n)$  & $\overline{1}$ & $\overline{1}$& $\overline{1}$&$\overline{1}$& $\overline{1}$& $\overline{4}$&$\overline{1}$&$\overline{6}$&$\overline{227}$&${191}$&18
 \end{tabular}}\]
\caption{ \label{numberup42} \rm\em Number of known regular two-graphs on up to 42 vertices}\end{table}
 \end{center}

Strongly regular graphs with at most $36$ vertices are fully classified and for the parameters $(37,18,8,9)$ all graphs with nontrivial automorphisms have been enumerated (see \cite{99, srg37}). The goal of this paper is to classify strongly regular graphs with parameters $(41,20,9,10)$ admitting nontrivial automorphisms and to enumerate regular two-graphs on $38$ and $42$ vertices with at least one descendant with a nontrivial automorphism.

The paper is organized as follows: After a brief description of the terminology and some background results in Section \ref{sec-bat},  we establish the existence of three new regular two-graphs on $38$ vertices in Section \ref{two38} and complete the classification of such two-graphs with at least one descendant with a nontrivial automorphism. We also construct $36$ new strongly regular graphs with parameters $(37,18,8,9)$ and a trivial automorphism group as descendants of the newly constructed two-graphs. In Section  \ref{srg41}, we apply the method for constructing strongly regular graphs using orbit matrices to construct all strongly regular graphs with parameters $(41,20,9,10)$ and nontrivial automorphism groups. Using this classification, in Section \ref{two42}, we construct all regular two-graphs on $42$ vertices with at least one descendant with a nontrivial automorphism group. Moreover, by constructing all descendants of the new two-graphs we obtain new  strongly regular graphs with parameters  $(41,20,9,10)$ and a trivial automorphism group. \\
To eliminate isomorphic graphs and to determine order and the structure of their automorphism groups we use GAP \cite{GAP}.

\section{Background and terminology}\label{sec-bat}

We assume that the reader is familiar with basic notions from the theory of finite groups. For basic definitions and properties of strongly regular graphs and two-graphs,
we refer the reader to \cite{ bro-ham, G-R, McKay-Spence, crc, vdt}.

\smallskip

A graph is {regular} if all its vertices have the same valency. A simple regular graph $\Gamma=(\mathcal{V},\mathcal{E})$ is
{strongly regular} with parameters $(v,k, \lambda, \mu )$ if it has $|\mathcal{V}|=v$ vertices, valency $k$,
and if any two adjacent vertices are together adjacent to $\lambda$ vertices, while any two nonadjacent vertices are together adjacent to $\mu$ vertices. A strongly regular graph  with parameters $(v,k, \lambda, \mu )$ is usually denoted by SRG$(v, k, \lambda, \mu)$.  A conference graph is a strongly regular graph with parameters $(v, k = (v -1)/2, \lambda = (v-5)/4, \mu= (v-1)/4)$.

An automorphism of a strongly regular graph $\Gamma$ is a permutation of the vertices of $\Gamma$, such that two vertices are adjacent if and only if their images are adjacent. The full automorphism group of $\Gamma$, usually denoted by $Aut(\Gamma)$, is the group of all such permutations.  Let $\Gamma_{1}=(\mathcal{V},\mathcal{E}_{1})$ and $\Gamma_{2}=(\mathcal{V},\mathcal{E}_{2})$ be strongly regular graphs and $G\leq Aut(\Gamma_1)\cap Aut(\Gamma_2)$.
An isomorphism $\alpha:\Gamma_1\rightarrow \Gamma_2$ is called a {$G$-isomorphism} if there exists an automorphism
$\tau : G \rightarrow G$ such that for every $x,y \in \mathcal{V}$ and every $g \in G$ the following holds:
\[(\tau g).(\alpha x)= \alpha y \Leftrightarrow g.x=y.\]



A two-graph is a pair $(\mathcal{V},\Delta)$, where  $\Delta$ is a collection of unordered triples chosen from a finite set of vertices $\mathcal{V}$, such that every $4$-subset of $\mathcal{V}$ contains an even number of triples of $\Delta$. The triples from $\Delta$ are called coherent. A regular two-graph has the property that every pair of vertices lies in the same number of triples of the two-graph. The complement of the two-graph $(\mathcal{V},\Delta)$ is the two-graph $(\mathcal{V},\overline{\Delta})$, where $\overline{\Delta}$ is the complement of ${\Delta}$ in the set of all $3$-subsets of $\mathcal{V}$. The two-graphs $(\mathcal{V},\Delta)$ and $(\mathcal{V}',\Delta')$ are isomorphic if there exists a bijection
$\mathcal{V} \rightarrow \mathcal{V}'$ that induces a bijection $\Delta \rightarrow \Delta'$. A two-graph is called self-complementary if it is isomorphic to its complement. The automorphism group  $Aut(\mathcal{V},\Delta)$ of a two-graph $(\mathcal{V},\Delta)$ is the group of permutations of $\mathcal{V}$ which preserves $\Delta$.


From a two-graph $\Phi=(\mathcal{V},\Delta)$ and any fixed $x\in \mathcal{V}$ we construct the graph $\Gamma$ which has a vertex set $\mathcal{V}$ by setting the vertex $x$ to be an  isolated vertex and letting any two other vertices $y, z$   be adjacent in $\Gamma$ if $\{z, x, y\}$ is coherent in $\Phi$. Deleting the isolated vertex $x$ yields a graph on $|\mathcal{V}|-1$ vertices, which is called the descendant of $\Phi$. The two-graph $(\mathcal{V},\Delta)$  is regular if and only if each descendant is strongly regular with parameters $(v-1, k, \lambda, \mu)$, where $\mu=k/2$. If the descendants are conference graphs, the corresponding two-graph is called a conference two-graph. The  complement  of  any  conference  two-graph  is again  a conference two-graph.


In this paper, we classify SRGs$(41,20,9,10)$ with nontrivial automorphisms. Note that these strongly regular graphs are conference graphs. We also enumerate regular two-graphs on $38$ and $42$ vertices with at least one descendant with nontrivial automorphisms, which are conference two-graphs.
More details on conference graphs can be found in \cite{bro-ham,crc} .



\section{Regular two-graphs on 38 vertices} \label{two38}

It is known that there are at least $191$ regular two-graphs on $38$ vertices (see \cite{McKay-Spence}). These two-graphs are available at \cite{spence2}. In total, they have $6760$ nonisomorphic descendants, which are strongly regular graphs with parameters $(37,18,8,9)$.

Recently, Crnkovi\'c and the first author classified strongly regular graphs with parameters $(37,18,8,9)$  admitting nontrivial automorphisms. They constructed six new strongly regular graphs with parameters $(37,18,8,9)$ whose full automorphism group is of order three and showed that there are exactly forty SRGs$(37,18,8,9)$ with nontrivial automorphisms (see \cite [Theorem 4.3]{srg37}). By analysing the new strongly regular graphs with parameters $(37,18,8,9)$ constructed in \cite{srg37}, we obtain the following theorem.

\begin{theorem} \label{th_two_38}
Up to isomorphism, there exist at least $194$ regular two-graphs on $38$ vertices with $6802$ nonisomorphic descendants. Among them, there are $64$ self-complementary two-graphs. Exactly $20$ regular two-graphs on $38$ vertices have at least one descendant admitting nontrivial automorphisms, and there are no more two-graphs with this property.
\end{theorem}

\begin{proof}
By constructing two-graphs corresponding to the six new  SRGs$(37,18,8,9)$ constructed in \cite{srg37} and eliminating isomorphic copies, we obtain three new two-graphs ${\Phi}_{i}$, $i \in \{1,2,3\}$. Further analysis of these two-graphs leads to the results presented in Table \ref{tabtwo-38}.
The second column of the table contains the order of the corresponding full automorphism group $G_{\Phi_i}$, and the third column gives the number of nonisomorphic descendants with a given full automorphism group, where $E$ denotes a trivial group. In the last column, we indicate whether a two-graph is self-complementary or not.
\begin{center}
 \begin{table}[htpb!]
 \[
\begin{tabular}{l|l|l|c}
  \hline
	i&$|G_{\Phi_i}|$ &\text{Descendants of   ${\Phi}_{i}$}& "S"
\cr \hline \hline
1&3&$[12\times E, 2\times Z_{3}]$&NO\cr
2&3&$[12\times E, 2\times Z_{3}]$&NO\cr
3&3&$[12\times E, 2\times Z_{3}]$&YES\cr
	\hline\hline
 \end{tabular}\]
\caption{ \label{tabtwo-38}\rm\em Descendants of the new two-graphs on $38$ vertices}\end{table}
 \end{center}

\vspace*{-0.7cm}
 The two-graph ${\Phi}_{3}$ is a self-complementary two-graph, and ${\Phi}_{1}$ and ${\Phi}_{2}$ are complements to each other.
Each of the two-graphs ${\Phi}_{i}$, $i \in \{1,2,3\}$, has $14$ mutually nonisomorphic descendants. Among them, there are $12$ strongly regular graphs with parameters $(37,18,8,9)$ with a trivial automorphism group and two descendants with the full automorphism group of order three. We have thus obtained $36$ descendants that are new SRGs$(37,18,8,9)$ whose full automorphism group is trivial. Together with the previously known results, this includes all two-graphs that can have descendants with a nontrivial automorphism and gives the statement of the theorem.
\end{proof}
From the analysis in the proof of Theorem \ref{th_two_38} and previously known results (see \cite [Theorem 4.3]{srg37}) we have the following statement.

\begin{theorem}
Up to isomorphism, there exist at least $6802$ strongly regular graphs with parameters $(37,18,8,9)$. These are exactly forty SRGs$(37,18,8,9)$ admitting nontrivial automorphisms, and at least $6762$ SRGs$(37,18,8,9)$ with the full automorphism group of
order one.
\end{theorem}

As a supplement to the two-graphs given in \cite{bms} (see also \cite{spence2}), we give new two-graphs  represented by the adjacency matrix of one of their descendants (${\Phi}_{2}$ can be obtained from ${\Phi}_{1}$ as
its complement). We denote by $AM_{\Phi_i}$ the adjacency matrix corresponding to the descendant of ${\Phi}_{i}$.

\begin{center}
 \begin{table}[htpb!]
 \[
\small\addtolength{\tabcolsep}{-5pt}
\begin{tabular}{ c }
$AM_{{\Phi}_{1}}=\left(\begin{smallmatrix}0&0&0&0&1&1&0&0&1&0&0&1&0&1&1&1&1&0&1&1&1&0&1&1&1&1&0&1&0&0&1&0&0&0&0&0&1\\
0&0&0&1&1&0&1&0&1&1&0&1&1&1&0&1&0&1&0&0&0&1&1&0&1&0&1&1&1&0&1&1&0&0&0&0&0\\
0&0&0&1&0&1&0&1&1&0&1&1&1&0&1&0&1&1&0&0&0&1&0&1&0&1&1&1&0&1&1&0&1&0&0&0&0\\
0&1&1&0&1&1&0&1&0&0&1&0&0&1&0&0&0&0&0&1&1&1&1&1&1&0&1&0&1&1&1&0&0&1&0&0&0\\
1&1&0&1&0&0&0&1&1&0&1&1&0&0&0&0&1&0&0&1&0&1&1&0&1&0&0&1&0&0&0&1&1&1&0&1&1\\
1&0&1&1&0&0&0&0&0&0&0&0&1&1&0&1&0&0&0&0&1&1&0&1&1&1&1&1&0&0&0&1&1&1&1&0&1\\
0&1&0&0&0&0&0&0&1&1&1&0&0&0&1&1&0&0&0&1&1&0&0&1&1&1&1&1&1&1&0&1&1&0&0&1&0\\
0&0&1&1&1&0&0&0&1&1&1&0&0&1&0&1&1&1&1&0&1&0&1&0&0&1&0&0&0&1&0&1&1&1&0&0&0\\
1&1&1&0&1&0&1&1&0&0&0&1&0&0&0&1&0&1&0&0&1&0&0&0&0&1&1&1&0&1&1&0&0&1&0&1&1\\
0&1&0&0&0&0&1&1&0&0&0&1&1&1&0&0&0&0&1&1&0&0&1&1&0&1&0&1&0&1&1&1&1&1&1&0&0\\
0&0&1&1&1&0&1&1&0&0&0&1&1&0&1&1&0&1&0&1&1&0&1&1&1&0&0&0&0&0&0&0&1&0&1&1&0\\
1&1&1&0&1&0&0&0&1&1&1&0&1&1&1&1&0&0&0&1&0&1&0&0&0&0&0&0&0&1&1&0&1&0&1&0&1\\
0&1&1&0&0&1&0&0&0&1&1&1&0&1&1&0&1&1&0&0&1&0&0&0&1&0&0&1&0&0&1&1&0&1&1&1&0\\
1&1&0&1&0&1&0&1&0&1&0&1&1&0&0&1&1&1&0&1&1&0&0&1&0&0&0&0&1&1&0&1&0&0&0&0&1\\
1&0&1&0&0&0&1&0&0&0&1&1&1&0&0&0&1&0&1&0&1&0&1&0&1&0&1&0&1&1&1&1&1&0&0&0&1\\
1&1&0&0&0&1&1&1&1&0&1&1&0&1&0&0&0&1&1&0&1&1&0&0&1&1&0&0&1&0&0&0&1&0&1&0&0\\
1&0&1&0&1&0&0&1&0&0&0&0&1&1&1&0&0&1&1&1&0&1&0&0&1&1&0&1&1&1&0&1&0&0&0&1&0\\
0&1&1&0&0&0&0&1&1&0&1&0&1&1&0&1&1&0&1&0&0&0&1&1&0&0&1&1&1&0&0&0&0&0&1&1&1\\
1&0&0&0&0&0&0&1&0&1&0&0&0&0&1&1&1&1&0&1&1&1&1&0&0&0&1&1&1&0&1&0&1&1&1&0&0\\
1&0&0&1&1&0&1&0&0&1&1&1&0&1&0&0&1&0&1&0&1&1&0&1&0&0&1&1&0&1&0&0&0&0&1&1&0\\
1&0&0&1&0&1&1&1&1&0&1&0&1&1&1&1&0&0&1&1&0&0&0&0&0&0&1&0&0&0&1&1&0&1&0&1&0\\
0&1&1&1&1&1&0&0&0&0&0&1&0&0&0&1&1&0&1&1&0&0&0&0&0&1&1&0&1&0&1&1&1&0&1&1&0\\
1&1&0&1&1&0&0&1&0&1&1&0&0&0&1&0&0&1&1&0&0&0&0&1&1&1&1&0&0&0&1&1&0&0&1&0&1\\
1&0&1&1&0&1&1&0&0&1&1&0&0&1&0&0&0&1&0&1&0&0&1&0&0&1&0&1&1&0&1&0&1&0&0&1&1\\
1&1&0&1&1&1&1&0&0&0&1&0&1&0&1&1&1&0&0&0&0&0&1&0&0&1&0&1&1&1&0&0&0&1&1&0&0\\
1&0&1&0&0&1&1&1&1&1&0&0&0&0&0&1&1&0&0&0&0&1&1&1&1&0&0&0&0&1&1&1&0&0&1&1&0\\
0&1&1&1&0&1&1&0&1&0&0&0&0&0&1&0&0&1&1&1&1&1&1&0&0&0&0&1&0&1&0&1&0&0&1&0&1\\
1&1&1&0&1&1&1&0&1&1&0&0&1&0&0&0&1&1&1&1&0&0&0&1&1&0&1&0&0&0&0&0&1&1&0&0&0\\
0&1&0&1&0&0&1&0&0&0&0&0&0&1&1&1&1&1&1&0&0&1&0&1&1&0&0&0&0&1&1&0&1&1&0&1&1\\
0&0&1&1&0&0&1&1&1&1&0&1&0&1&1&0&1&0&0&1&0&0&0&0&1&1&1&0&1&0&0&0&0&1&1&0&1\\
1&1&1&1&0&0&0&0&1&1&0&1&1&0&1&0&0&0&1&0&1&1&1&1&0&1&0&0&1&0&0&0&0&1&0&1&0\\
0&1&0&0&1&1&1&1&0&1&0&0&1&1&1&0&1&0&0&0&1&1&1&0&0&1&1&0&0&0&0&0&1&0&0&1&1\\
0&0&1&0&1&1&1&1&0&1&1&1&0&0&1&1&0&0&1&0&0&1&0&1&0&0&0&1&1&0&0&1&0&1&0&0&1\\
0&0&0&1&1&1&0&1&1&1&0&0&1&0&0&0&0&0&1&0&1&0&0&0&1&0&0&1&1&1&1&0&1&0&1&1&1\\
0&0&0&0&0&1&0&0&0&1&1&1&1&0&0&1&0&1&1&1&0&1&1&0&1&1&1&0&0&1&0&0&0&1&0&1&1\\
0&0&0&0&1&0&1&0&1&0&1&0&1&0&0&0&1&1&0&1&1&1&0&1&0&1&0&0&1&0&1&1&0&1&1&0&1\\
1&0&0&0&1&1&0&0&1&0&0&1&0&1&1&0&0&1&0&0&0&0&1&1&0&0&1&0&1&1&0&1&1&1&1&1&0 \end{smallmatrix}\right)$ \\
\\
$AM_{{\Phi}_{3}}=\left(\begin{smallmatrix}0&0&0&0&1&1&0&0&1&0&0&1&0&1&1&1&1&0&1&1&1&0&1&1&1&1&0&1&0&0&1&0&0&0&0&0&1\\
0&0&0&1&1&0&1&0&1&1&0&1&1&1&0&1&0&1&0&0&0&1&1&0&1&0&1&1&1&0&1&1&0&0&0&0&0\\
0&0&0&1&0&1&0&1&1&0&1&1&1&0&1&0&1&1&0&0&0&1&0&1&0&1&1&1&0&1&1&0&1&0&0&0&0\\
0&1&1&0&1&1&0&1&0&1&0&0&1&1&1&0&0&0&0&1&1&0&0&1&1&0&1&0&1&1&1&0&0&1&0&0&0\\
1&1&0&1&0&0&0&1&1&0&0&0&1&1&0&0&1&0&0&1&0&1&0&1&1&0&0&1&0&0&0&1&1&1&0&1&1\\
1&0&1&1&0&0&0&0&0&1&0&1&1&0&1&1&0&0&0&0&1&0&0&0&1&1&1&1&0&0&0&1&1&1&1&0&1\\
0&1&0&0&0&0&0&0&1&1&1&0&0&0&1&1&0&0&0&1&1&1&1&1&1&1&1&0&0&1&0&1&1&0&0&1&0\\
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1&1&1&0&0&1&0&0&1&1&1&0&0&1&0&0&1&0&0&1&0&1&0&0&0&1&1&0&1&0&1&1&0&0&1&0&1\\
0&1&1&1&1&1&0&0&0&0&1&0&0&1&1&1&0&1&0&1&0&1&1&0&0&1&0&1&0&0&0&0&0&1&1&1&0\\
1&1&0&1&1&0&0&1&0&0&1&1&1&0&0&1&0&0&1&1&0&0&1&0&0&1&1&0&1&1&0&0&1&0&0&0&1\\
1&0&1&1&0&1&1&0&0&0&0&0&1&0&0&1&1&1&0&1&1&1&1&1&0&0&0&0&1&1&0&1&0&0&0&0&1\\
1&1&0&0&0&1&1&1&1&0&0&0&1&1&1&0&0&1&1&0&1&1&0&0&1&1&0&0&1&0&0&0&1&0&1&0&0\\
1&0&1&0&1&0&0&1&0&1&0&1&0&0&1&0&0&1&1&1&0&1&0&0&1&1&0&1&1&1&0&1&0&0&0&1&0\\
0&1&1&0&0&0&0&1&1&1&0&0&1&0&1&1&1&0&1&0&0&0&1&1&0&0&1&1&1&0&0&0&0&0&1&1&1\\
1&0&0&0&0&0&0&1&0&1&0&0&0&1&0&1&1&1&0&1&1&1&1&0&0&0&1&1&0&1&1&0&1&1&1&0&0\\
1&0&0&1&1&0&1&0&0&1&1&1&1&1&1&0&1&0&1&0&1&1&0&1&0&0&1&0&0&0&0&0&0&0&1&1&0\\
1&0&0&1&0&1&1&1&1&0&1&0&0&0&1&1&0&0&1&1&0&0&0&0&0&0&1&1&1&0&1&1&0&1&0&1&0\\
0&1&1&0&1&0&1&0&1&0&0&1&1&0&1&1&1&0&1&1&0&0&0&0&0&0&0&0&0&1&1&1&1&1&1&0&0\\
1&1&0&0&0&0&1&0&0&1&1&0&1&1&1&0&0&1&1&0&0&0&0&1&0&1&0&1&0&1&1&1&0&1&0&0&1\\
1&0&1&1&1&0&1&1&1&1&1&0&0&0&1&0&0&1&0&1&0&0&1&0&1&0&0&0&0&0&1&0&1&0&1&0&1\\
1&1&0&1&1&1&1&0&0&1&0&0&0&0&0&1&1&0&0&0&0&0&0&1&0&1&0&1&1&1&1&0&1&0&1&1&0\\
1&0&1&0&0&1&1&1&1&1&1&1&1&1&0&1&1&0&0&0&0&0&1&0&1&0&0&0&0&1&0&0&0&1&0&1&0\\
0&1&1&1&0&1&1&0&1&1&0&1&0&1&0&0&0&1&1&1&1&0&0&0&0&0&0&1&0&1&0&0&1&0&0&1&1\\
1&1&1&0&1&1&0&0&0&0&1&0&1&0&0&0&1&1&1&0&1&0&1&0&1&0&1&0&0&0&1&1&1&0&0&1&0\\
0&1&0&1&0&0&0&1&0&0&1&1&0&1&1&1&1&1&0&0&1&0&0&0&1&0&0&0&0&1&1&1&0&0&1&1&1\\
0&0&1&1&0&0&1&0&0&0&0&0&0&1&1&0&1&0&1&0&0&1&1&0&1&1&1&0&1&0&1&0&1&1&0&1&1\\
1&1&1&1&0&0&0&0&1&0&1&1&0&0&0&0&0&0&1&0&1&1&1&1&1&0&0&1&1&1&0&0&0&1&1&0&0\\
0&1&0&0&1&1&1&1&0&1&1&1&0&0&1&0&1&0&0&0&1&1&1&0&0&0&0&1&1&0&0&0&1&1&0&0&1\\
0&0&1&0&1&1&1&1&0&0&1&0&0&1&0&1&0&0&1&0&0&1&0&1&1&0&1&1&0&1&0&1&0&0&1&0&1\\
0&0&0&1&1&1&0&1&1&1&0&0&1&0&0&0&0&0&1&0&1&1&1&0&0&1&0&0&0&1&1&1&0&0&1&1&1\\
0&0&0&0&0&1&0&0&0&1&1&1&1&0&0&1&0&1&1&1&0&1&0&1&1&0&0&0&1&0&1&0&1&1&0&1&1\\
0&0&0&0&1&0&1&0&1&0&1&0&1&0&0&0&1&1&0&1&1&0&0&0&1&1&1&1&1&1&0&0&0&1&1&0&1\\
1&0&0&0&1&1&0&0&1&0&0&1&0&1&1&0&0&1&0&0&0&0&1&1&0&0&1&0&1&1&0&1&1&1&1&1&0 \end{smallmatrix}\right) $\\
\end{tabular}\]
\end{table}
 \end{center}






\section{ Enumeration of SRGs($41,20,9,10$) with nontrivial automorphisms} \label{srg41}

There are 120 mutually nonisomorphic SRGs$(41,20,9,10)$ arising  as  descendants  of  18  regular two-graphs $\Phi_i$, $1\leq i \leq 18$, on $42$ vertices constructed by Bussemaker, Mathon and Seidel in \cite{bms}. These graphs are presented in Table \ref{tabtwo-z3-spence}, where we use the same notation as in Table \ref{tabtwo-38}. In Table \ref{tabtwo-z3-spence}, a pair of complementary two-graphs is represented by one of them.

\begin{center}
 \begin{table}[h]%[htpb!]
 \[ { \begin{tabular}{l|l|l|c}
  \hline
	i&$|{\rm Aut}({\Phi}_{i})|$ &\text{Descendants of} ${\Phi}_{i}$& "S"
\cr \hline \hline
1&34440&$[1\times Z_{41}: Z_{30}]$&YES\cr
2&168&$[1\times Z_{4}]$&YES\cr
3&21&$[1\times E]$&YES\cr
4&14&$[2\times E, 2\times Z_2]$&YES\cr
5&14&$[2\times E, 2\times Z_2]$&YES\cr
6&14&$[2\times E, 2\times Z_2]$&NO\cr
7&14&$[6\times Z_2]$&YES\cr
8&8&$[2\times E, 6\times Z_2, 2\times D_8]$&YES\cr
9&8&$[2\times E, 6\times Z_2, 2\times D_8]$&YES\cr
10&8&$[2\times E, 6\times Z_2, 2\times D_8]$&NO\cr
11&7&$[6\times E]$&YES\cr
12&7&$[6\times E]$&YES\cr
13&6&$[4\times E, 6\times Z_2]$&YES\cr
14&6&$[4\times E, 6\times Z_2]$&YES\cr
15&5&$[8\times E, 2\times Z_5]$&YES\cr
16&4&$[10\times E, 2\times Z_4]$&YES\cr
	\hline\hline
 \end{tabular}}\]
\caption{ \label{tabtwo-z3-spence}\rm\em Descendants of the known two-graphs on $42$ vertices}\end{table}
 \end{center}

 \vspace*{-0.8cm}
 Further, there are 80 strongly regular graphs with parameters $(41,20,9,10)$ having the full automorphism group isomorphic to the symmetric group $S_3$ (see \cite[Theorem 5] {50s3}). To the best of our knowledge, these $200$ graphs are the only known SRGs$(41,20,9,10)$. The aim of this section is to construct all SRGs$(41,20,9,10)$ with nontrivial automorphisms.\\

Let $G$ be an automorphism group of the graph $\Gamma$ with $|V|=v$ vertices, partitioning the set of vertices $V$ into $b$ orbits of sizes $n_1, \ldots , n_b$, respectively, where $\sum_{i=1}^b n_i=v$.
It is known that $n_i$ divides $|G|$, for $i=1, \ldots, b $. Thus, to enumerate all strongly regular graphs with parameters $(41,20,9,10)$ admitting nontrivial automorphisms, we consider automorphisms of prime order, following the construction method proposed by Behbahani and Lam in \cite{majid-lam}. They introduced the concept of orbit matrices of strongly regular graphs and gave a method for constructing orbit matrices of strongly regular graphs with automorphisms of prime order and corresponding strongly regular graphs (see \cite{beh, majid-lam}). In our construction, we will use the column orbit matrices introduced in \cite{amc}.

 \begin{definition}
 A $(b \times b)$-matrix $C = [c_{ij}]$ with entries satisfying conditions:
 \begin{align}
 \sum_{i=1}^b c_{ij}&=\sum_{j=1}^b \frac{n_j}{n_i} c_{ij}=k,  \label{s3}\\
 \sum_{s=1}^b  \frac{n_s}{n_j} c_{is} c_{js}&= \delta_{ij} (k- \mu ) + \mu n_i + ( \lambda - \mu) c_{ij}, \label{s4}
 \end{align}
 where  $0\leq c_{ij}\leq n_i$, $0\leq c_{ii}\leq n_i-1$ and $\sum_{i=1}^{b}n_i=v$, is called a {\bf column orbit matrix}
 for a strongly regular graph
  with parameters $(v,k, \lambda, \mu)$ and the orbit lengths distribution $(n_1, \ldots, n_b)$.
 \end{definition}


There is exactly one SRG$(41,20,9,10)$ admitting an automorphism group isomorphic to $Z_{41}$, namely the Paley graph with $41$ vertices having the full automorphism group isomorphic to $Z_{41}:Z_{30}$. In the sequel, we will consider automorphisms of prime order $p$, where $2 \leq p \leq 37$. Such automorphism acts in orbits of at most two different lengths. If the group $G$ acts with $d_1$ orbits of length $1$ and $d_p$ orbits of length $p$, we denote the corresponding orbit lengths distribution by $(d_1\times 1, d_p\times p)$.

\begin{lemma}
If an automorphism of prime order $p$, $2\leq p\leq 37,$ acts on a strongly regular graph with parameters $(41,20,9,10)$, then $p \in \{2, 3, 5 \}$.
\end{lemma}

\begin{proof}
The first step in constructing strongly regular graphs with parameters\linebreak  $(41,20,9,10)$ admitting an automorphism of prime order $p$ is to determine all permissible distributions $(d_1 \times 1, d_p \times p)$, $2 \leq p \leq 37$. A nontrivial automorphism acting on SRG$(v,k,\lambda,\mu)$ with eigenvalues $s<r<k$ fixes at most $ \frac{\max(\lambda,\mu)}{k-r}v$ vertices (\cite[Theorem 3.7]{beh}). Therefore, $d_1\leq 23$, $d_1+d_p\cdot p =41$, and the number of possible orbit lenghts  distributions is as given in the second row of Table \ref{orb_distr}.

\vspace*{-0.3cm}
\begin{center} \footnotesize
\begin{table}[h]%[htpb!]
\[ { \begin{tabular}{c||c|c|c|c|c|c|c|c|c|c|c|c}
  \hline
$p$&2&3&5&7&11&13&17&19&23&29&31&37\\
\hline
No. of distributions&12&8&5&3&2&2&1&2&1&1&1&1\\
\hline
Distributions with&10 & 2& 2&0&0&0&0&0&0&0&0&0\\
prototypes& & & & & & & & & & & &
  \cr \hline 	
 \end{tabular}}\]
\caption{\label{orb_distr} \rm \em The number of orbit lenghts    distributions and the existence of prototypes}\end{table}
 \end{center}

\vspace*{-0.7cm}
In \cite{beh}, the concept of a prototype for a row of a column orbit matrix $C=[c_{ij}]$ of a strongly regular graph with a presumed automorphism group of prime order $p$ was  introduced.

A prototype of a fixed row (a row corresponding to an orbit of length 1) for the distribution $(d_1\times 1, d_p\times p)$ is a nonnegative integer solution of $x_0, x_1, y_0$ and $y_1$ satisfying the following set of linear equations:
\begin{equation} \label{eq1}
\begin{gathered}
x_0+x_1=d_1 \\
y_0+y_1=d_p \\
x_1+py_1=k, \\
\end{gathered}
\end{equation}
where $x_0$ and $x_1$ are the number of zeros and ones, respectively, in the fixed columns of a fixed row, and $y_0$ and $y_1$ are the number of zeros and ones, respectively, in the nonfixed columns of a fixed row.

A prototype of a nonfixed row (a row corresponding to an orbit of length $p$) for the distribution  $(d_1\times 1, d_p\times p)$ is a nonnegative integer solution of $x_0, x_p, y_0,\dots, y_p$, satisfying this set of linear equations:
\begin{equation} \label{eq2}
\begin{gathered}
x_0+x_p=d_1 \\
y_0+y_1+\dots+y_p=d_p \\
x_p+y_1+2y_2\dots+py_p=k \\
px_p+y_1+4y_2\dots+p^2y_p=\frac{(k-\mu)p +\mu p^2+(\lambda-\mu)c_{rr}p}{p}\\
\end{gathered}
\end{equation}
for any nonfixed row $r$, where $x_0$ and $x_p$ are the number of zeros and $p'$s, respectively, in the fixed columns of the row $r$,  and $y_i,  i=0,1,\dots,p,$ is the number of $i$'s on the nonfixed columns of the row $r$. So, for different $c_{rr}$ we get different equations. Since $p$ is a prime, the number $c_{rr}$ must be even (\cite[Lemma 3.2]{beh}).

Solving the systems of equations (\ref{eq1}) and (\ref{eq2}) for all possible orbit lengths distributions from Table \ref{orb_distr}, we obtain that prototypes exist only when $p \in \{2,3,5\},$ as presented in the third row of  Table \ref{orb_distr}. (The orbit lengths distributions for which row prototypes exist are given in the first three rows of Table \ref{srg}.)
\end{proof}


After eliminating orbit lengths distributions for which there are no row prototypes, we must consider the orbit lengths distributions $(d_1\times 1, d_p\times p)$ shown in Table \ref{srg}. Using the prototypes, we construct the orbit matrices row by row,  eliminating mutually $G$-isomorphic orbit matrices. For eliminating orbit matrices yielding $G$-isomorphic strongly regular graphs we use the same method as for eliminating orbit matrices of $G$-isomorphic designs (see \cite{c-r-metrika,50s3}). The construction was performed using our programs written in GAP \cite{GAP}. The number of constructed orbit matrices is listed in Table \ref{srg}.

\vspace*{-0.3cm}
\begin{center} \footnotesize
\begin{table}[h]%[htpb!]
\[ { \begin{tabular}{c||c|c|c|c|c|c|c|c|c|c|c|c|c|c}
  \hline
$p$&2&2&2&2&2&2&2&2&2&2&3&3&5&5\\
\hline
$d_1$&1&3&5&7&9&11&13&15&17&19&5&8&1&6\\
\hline
$d_p$&20 & 19& 18&17&16&15&14&13&12&11&12&11&8&7
  \cr \hline
$\# OM$&6 & 0& 2872&0&6&0&232&0&0&0&18&0&3&2
  \cr \hline
$\# SRG$&12 & 0& 2362&0&64&0&4544&0&0&0&264&0&3&0
  \cr \hline
 \end{tabular}}\]
\caption{ \label{srg} \rm\em The number of constructed orbit matrices and SRGs($41,20,9,10$)}\end{table}
 \end{center}

\vspace*{-0.7cm}
Orbit matrices exist for the orbit lenghts distributions $(1\times 1, 20 \times 2)$, $(5\times 1, 18 \times 2)$, $(9\times 1, 16 \times 2)$, $(13\times 1, 14\times 2)$, $(5\times 1, 12\times 3)$, $(1\times 1, 8 \times 5)$ and $(6\times 1, 7\times 5)$, and in the final step of the construction, we consider these cases to construct adjacency matrices of  strongly regular graphs with parameters $(41,20,9,10)$. The number of nonisomorphic graphs we obtain is given in the fifth row of   Table \ref{srg}. Among the constructed strongly regular graphs, there are $7152$ mutually nonisomorphic graphs, and $7089$ of them are new. An analysis of the full automorphism groups of these $7152$ graphs gives us the following theorem.

\begin{theorem}
Up to isomorphism, there exist  exactly $7152$ strongly regular graphs with parameters $(41,21,9,10)$ having nontrivial automorphisms, with the full automorphism groups as presented in Table \ref{srg_41}.
\end{theorem}
\begin{center} \footnotesize
\begin{table}[h]%[htpb!]
\[ { \begin{tabular}{c||c|c|c|c|c|c|c}
  \hline
${\rm Aut}({\Gamma})$ &$Z_{41}:Z_{30}$&$D_8$&$S_3$&$Z_5$&$Z_4$&$Z_3$&$Z_2$\\
\hline
$|{\rm Aut}({\Gamma})|$ &820&8&6&5&4&3&2\\
\hline
$  \#SRGs $&1 & 8& 80&2&3&184&6874
  \cr \hline
	
 \end{tabular}}\]
\caption{\label{srg_41} \rm\em SRGs$(41,20,9,10)$ with nontrivial automorphisms}\end{table}
 \end{center}

\vspace*{-0.7cm}
The list of adjacency matrices of all SRGs$(41,20,9,10)$ with nontrivial automorphisms  is available online at:  \begin{verbatim}
 http://www.math.uniri.hr/~mmaksimovic/nontrivial_41.txt .
 \end{verbatim}

\section{Regular two-graphs on 42 vertices} \label{two42}

In \cite{bms}, Bussemaker, Mathon and Seidel constructed $18$ regular two-graphs on $42$ vertices (see Table \ref{tabtwo-z3-spence}). To the best of our knowledge, these are the only known regular two-graphs on $42$ vertices. \\

\begin{theorem} \label{th_two42}
Up to isomorphism, there exist at least $752$ regular two-graphs on $42$ vertices with $18439$ nonisomorphic descendants. Among them, there are $64$ self-complementary two-graphs. Exactly $749$ regular two-graphs on $42$ vertices have at least one descendant with a nontrivial automorphism, and there are no more two-graphs with this property.
\end{theorem}


\begin{proof}
Using the classification of SRGs($41,20,9,10$) with nontrivial automorphisms given in Section \ref{srg41}, we constructed all   corresponding two-graphs on $42$ vertices and their descendants. (Note that if there are more two-graphs on $42$ vertices, they can only have descendants with a trivial automorphism group.) Our results are summarized in Table \ref{tabtwo}.
\begin{center}
 \begin{table}[htpb!]
 \[ { \begin{tabular}{l|l|l|c|c}
  \hline
	i&$|{\rm Aut}({\Phi}_{i})|$ &\text{Descendants of} ${\Phi}_{i}$& "S"&New
\cr \hline \hline
1&34440&$[1\times Z_{41}: Z_{30}]$&YES&NO\cr
2&168&$[1\times Z_{4}]$&YES&NO\cr
3-4&14&$[2\times E, 2\times Z_2]$&YES&NO\cr
5&14&$[2\times E, 2\times Z_2]$&NO&NO\cr
6&14&$[6\times Z_2]$&YES&NO\cr
7&9&$[4\times E, 2\times Z_3]$&YES&YES\cr
8-9&8&$[2\times E, 6\times Z_2, 2\times D_8]$&YES&NO\cr
10&8&$[2\times E, 6\times Z_2, 2\times D_8]$&NO&NO\cr
11-12&6&$[4\times E, 6\times Z_2]$&YES&NO\cr
13-32&6&$[4\times E, 4\times Z_2, 2\times Z_3, 2\times S_3]$&NO&YES\cr
33&5&$[8\times E, 2\times Z_5]$&YES&NO\cr
34&4&$[10\times E, 2\times Z_4]$&YES&NO\cr
35&3&$[12\times E, 6\times Z_3]$&YES&YES\cr
36-43&3&$[12\times E, 6\times Z_3]$&NO&YES\cr
44-63&2&$[18\times E, 6\times Z_2]$&YES&YES\cr
64-227&2&$[18\times E, 6\times Z_2]$&NO&YES\cr
228-253&2&$[14\times E, 14\times Z_2]$&YES&YES\cr
254-402&2&$[14\times E, 14\times Z_2]$&NO&YES\cr
403-404&2&$[16\times E, 10\times Z_2]$&YES&YES\cr
405&2&$[16\times E, 10\times Z_2]$&NO&YES\cr
	\hline\hline
 \end{tabular}}\]
\caption{ \label{tabtwo} \rm\em Descendants of the constructed two-graphs on $42$ vertices}\end{table}
 \end{center}

\vspace*{-0.8cm}

Among the $749$ constructed two-graphs, there are $61$ self-complementary two-graphs and $344$ pairs of complementary two-graphs. In total, these $749$ two-graphs have  $18426$ descendants. In Table \ref{tabtwo}, each pair of complementary two-graphs is represented by one of them.


 The only known two-graphs on $42$ vertices not included in Table \ref{tabtwo} are those which have only descendants with a trivial automorphism group; there are three such two-graphs known (see \cite{bms}), and together they have $13$ nonisomorphic descendants with a trivial automorphism group (see Table \ref{tabtwo-z3-spence}). So, the statement of the theorem holds.
\end{proof}

For each two-graph from Table \ref{tabtwo} the adjacency matrix of one of its descendants with the smallest nontrivial automorphism group is available online at:  \begin{verbatim}
  http://www.math.uniri.hr/~mmaksimovic/descendants_41.txt .
  \end{verbatim}
Up to isomorphism, all two-graphs on $42$ vertices with at least one descendant with nontrivial automorphism group can be reconstructed from this list, as can their $18426$ descendants.\\

As can be seen from Table \ref{tabtwo-z3-spence}, the number of SRGs$(41,20,9,10)$ with a trivial full automorphism group, which have been known so far is $55$. Newly constructed two-graphs on $42$ vertices have together $11274$ nonisomorphic descendants whose full automorphism group is trivial and among them, there are $11232$ new SRGs$(41,20,9,10)$. Therefore, the following theorem holds.

\begin{theorem} \label{srg41_trivial}
Up to isomorphism, there exist at least  $18439$ strongly regular graphs with parameters $(41,20,9,10)$. There are exactly $7152$ strongly regular graphs with parameters $(41,20,9,10)$ having nontrivial automorphisms, and at least $11287$ strongly regular graphs with parameters $(41,20,9,10)$ having the full automorphism group of order one.
\end{theorem}

\begin{remark}
Strongly regular configurations with parameters $(41_5,9,10)$  are the smallest strongly regular configurations  for which (non)existence is not known (see \cite{src}). Such a configuration could arise from a strongly regular graph having parameters $(41,20,9,10)$. However, we have checked all  $18439$ strongly regular graphs from Theorem \ref{srg41_trivial} and none of them yields a strongly regular configuration. Thus, if a strongly regular configuration $(41_5,9,10)$ could be constructed from a strongly regular graph with parameters $(41,20,9,10)$, such a graph has no nontrivial automorphisms.
\end{remark}




\section*{Acknowledgement}

This work has been fully supported by the {\rm C}roatian Science Foundation under   project 6732.  The authors would like to thank the anonymous referee for valuable comments that improved the presentation of the paper.

\bigskip

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