On Some Regular Two-Graphs up to 50 Vertices

Abstract

Regular two-graphs on up to 36 vertices are classified, and recently, the classification of regular two-graphs on 38 and 42 vertices having at least one descendant with a nontrivial automorphism group has been performed. The first unclassified cases are those on 46 and 50 vertices. It is known that there are at least 97 regular two-graphs on 46 vertices leading to 2104 descendants and 54 regular two-graphs on 50 vertices leading to 785 descendants. In this paper, we classified all strongly regular graphs with parameters $(45,22,10,11)$, $(49,24,11,12)$, and $(50,21,8,9)$ that have $Z_6$ as the automorphism group and constructed regular two-graphs from SRGs $(45,22,10,11)$, SRGs $(49,24,11,12)$, and SRGs $(50,21,8,9)$ that have automorphisms of order six. In this way, we enumerated all regular two-graphs on up to 50 vertices that have at least one descendant with an automorphism group of order six or at least one strongly regular graph associated with an automorphism group of order six. We found 236 new regular two-graphs on 46 vertices leading to 3172 new SRG $(45,22,10,11)$ and 51 new regular two-graphs on 50 vertices leading to 398 new SRG $(49,24,11,12)$.

1. Introduction

A regular two-graph is the pair $\Phi =(\mathcal{V},\Delta )$, where $\mathcal{V}$ is a finite set of vertices and $\Delta$ is a set of unordered triples of $\mathcal{V}$ such that every four-subset of $\mathcal{V}$ contains an even number of triples of $\Delta$ and every pair of vertices lies in the same number of triples of $\Delta$. G. Higman introduced regular two-graphs, and his work was continued by Taylor in [1], where he connected regular two-graphs with strongly regular graphs. The first attempt at classifying regular two-graphs on at most 50 vertices was made by Bussemaker, Mathon, and Seidel in [2], and they produced a classification for all regular two-graphs with less than 30 vertices. Spence and McKay continued their work in [3,4]. In [5] also, the authors increased the number of known regular two-graphs on 38 and 42 vertices. As far as we know, the number of known regular two-graphs on up to 50 vertices is listed in

Table 1. Known regular two-graphs on up to 50 vertices.

n	$\mathit{N}\left(\mathit{n}\right)$	n	$\mathit{N}\left(\mathit{n}\right)$
6	$\overline{1}$	30	$\overline{6}$
10	$\overline{1}$	36	$\overline{227}$
14	$\overline{1}$	38	194
16	$\overline{1}$	42	752
18	$\overline{1}$	46	97
26	$\overline{4}$	50	54
28	$\overline{1}$

. In the table, n denotes the number of vertices and $N\left(n\right)$ is the number of known regular two-graphs, where the number is overlined if the classification is complete for the particular case.

The classification and enumeration of regular two-graphs is closely related to one of the main problems of strongly regular graph theory—the construction and classification of strongly regular graphs with given parameters.

Let $\Gamma =(\mathcal{V},\mathcal{E})$ be a simple regular graph with v vertices and of valency k. $\Gamma$ is a strongly regular graph with parameters $(v,k,\lambda ,\mu )$ if any two adjacent vertices have $\lambda$ common neighbors, while any two nonadjacent vertices have $\mu$ common neighbors. We denote strongly regular graphs with parameters $(v,k,\lambda ,\mu )$ by SRG $(v,k,\lambda ,\mu )$.

Regular two-graphs are related to strongly regular graphs in a few ways. First, the descendants of regular two-graph on v vertices are strongly regular graphs with parameters $(v-1,k,\lambda ,k/2)$. Moreover, a strongly regular graph with parameters $(v,k,\lambda ,\mu )$ will be associated with a regular two-graph if and only if $v=2(2k-\lambda -\mu )$. For more information about descendants and graphs associated with a two-graph, see Section 1.1.

Classification for strongly regular graphs with up to 36 vertices has been performed. The SRGs with up to 50 vertices that still need to be classified are those with parameters $(37,18,8,9)$, $(41,20,9,10)$, $(45,22,10,11)$, $(49,24,11,12)$, $(49,18,7,6)$, and $(50,21,8,9)$. In this work, we are interested in SRGs connected to regular two-graphs on up to 50 vertices, regardless of whether they are descendants or associated with a regular two-graph. Since SRGs $(37,18,8,9)$ and SRGs $(41,20,9,10)$ with nontrivial automorphisms were enumerated (see [5,6,7]), our interest has been on strongly regular graphs with parameters $(45,22,10,11)$, $(49,24,11,12)$, and $(50,21,8,9)$. The descendants of a regular two-graph on 46 vertices are SRGs $(45,22,10,11)$; the descendants of a regular two-graph on 50 vertices are SRGs $(49,24,11,12)$; for SRGs $(50,21,8,9)$, the associated two-graph on 50 vertices is regular.

In [8], we presented a method to construct strongly regular graphs with an automorphism group of composite order from orbit matrices. We used this method in this work to construct strongly regular graphs. Next, we constructed related regular two-graphs, and in a final step, we obtained descendants of these regular two-graphs.

The aim of this work was to enumerate SRGs $(45,22,10,11)$, SRGs $(49,24,11,12)$, and SRGs $(50,21,8,9)$ with $Z_6$ as the automorphism group and update the results on regular two-graphs on 46 and 50 vertices and their descendants.

Since the SRGs with parameters $(45,22,10,11)$, $(49,24,11,12)$, and $(50,21,8,9)$ having $S_3$ as the automorphism group are classified (see [9]), we enumerated SRGs $(45,22,10,11)$, SRGs $(49,24,11,12)$, and SRGs $(50,21,8,9)$ with an automorphism group of order six and constructed two-graphs related to them. In this way, we enumerated all regular two-graphs on 46 vertices and on 50 vertices that have at least one descendant with an automorphism group of order six or at least one strongly regular graph, associated with it having an automorphism group of order six. Thus, we found 236 new regular two-graphs on 46 vertices and 51 new regular two-graphs on 50 vertices, and we obtained 3172 new SRG $(45,22,10,11)$ and 398 new SRG $(49,24,11,12)$ as descendants of the corresponding regular two-graphs.

In Section 1.1, we give a brief overview of the basic definitions and background results. Then, in Section 2, we explain the methods we used in this work: construction of strongly regular graphs from their orbit matrices, construction of two-graphs from strongly regular graphs, and construction of descendants of two-graphs. We used the method of constructing strongly regular graphs with an automorphism group of composite order from their orbit matrices in Section 3 to classify strongly regular graphs with parameters $(45,22,10,11)$, $(49,24,11,12)$, and $(50,21,8,9)$ that have an abelian group of order six as the automorphism group. In addition to this classification, in Section 3.4, we use the classification of SRGs $(45,22,10,11)$ with a nonabelian group of order six as the automorphism group to construct all regular two-graphs on 46 that have at least one descendant with an automorphism of order six. In Section 3.5, we use the classification of SRGs $(49,24,11,12)$ and SRGs $(50,21,8,9)$ having an automorphism of order six to construct all regular two-graphs on 50 vertices that have at least one descendant with an automorphism of order six or at least one strongly regular graph with an automorphism of order six associated with it. We also construct descendants of the new regular two-graphs and obtain new strongly regular graphs with parameters $(45,22,10,11)$ and $(49,24,11,12)$. Finally, we give a brief overview and the conclusion in Section 4.

For the construction and study of the orbit matrices, graphs, and two-graphs, we used our programs written for GAP [10].

1.1. Basic Definitions and Background

Here, we give a brief review of the basic definitions and background results taken from [5,11,12,13,14]. For more details on strongly regular graphs and regular two-graphs, we refer the reader to [11,12,13,14].

A two-graph is the pair $\Phi =(\mathcal{V},\Delta )$, where $\mathcal{V}$ is a finite set of vertices and $\Delta$ is a set of unordered triples of $\mathcal{V}$, called coherent triples, with the property that every four-subset of $\mathcal{V}$ contains an even number of coherent triples. A regular two-graph is a two-graph where each pair of vertices is contained in the same number of coherent triples. The complement of two-graph $\Phi =(\mathcal{V},\Delta )$ is two-graph $\overline{\Phi }(\mathcal{V},\overline{\Delta })$ such that $\overline{\Delta }$ is the complement of $\Delta$ in the set of all three subsets of $\mathcal{V}$. A self-complementary two-graph is a two-graph that is isomorphic to its complement. We say that two-graphs $\Phi =(\mathcal{V},\Delta )$ and ${\Phi }^{\prime }=({\mathcal{V}}^{\prime },{\Delta }^{\prime })$ are isomorphic if there is a bijection $\mathcal{V}\to {\mathcal{V}}^{\prime }$ on the set of vertices that induces a bijection $\Delta \to {\Delta }^{\prime }$ on the set of coherent triples. The automorphism group of a two-graph is the group of permutations on the set of vertices that preserves the set of triples, and the full automorphism group is the group of all such permutations. For the two-graph $\Phi$, we denote its full automorphism group by $Aut(\Phi )$.

Let $\Gamma =(\mathcal{V},\mathcal{E})$ be a simple regular graph with v vertices and of valency k. $\Gamma$ is a strongly regular graph with parameters $(v,k,\lambda ,\mu )$ if any two adjacent vertices have $\lambda$ common neighbours, while any two nonadjacent vertices have $\mu$ common neighbours. We denote strongly regular graphs with parameters $(v,k,\lambda ,\mu )$ by SRG $(v,k,\lambda ,\mu )$. A conference graph is SRG $(v,(v-1)/2,(v-5)/4,(v-1)/4)$. An automorphism of a strongly regular graph is a permutation of the set of vertices preserving adjacency, and the full automorphism group is the group of all such permutations. For the graph $\Gamma$, we denote its full automorphism group by $Aut(\Gamma )$. For strongly regular graphs ${\Gamma }_1=(\mathcal{V},{\mathcal{E}}_1)$ and ${\Gamma }_2=(\mathcal{V},{\mathcal{E}}_2)$, and for $G\le Aut\left({\Gamma }_1\right)\cap Aut\left({\Gamma }_2\right)$, the G-isomorphism is an isomorphism $\alpha :{\Gamma }_1\to {\Gamma }_2$ for which there is an automorphism $\tau :G\to G$ such that, for every $x,y\in \mathcal{V}$ and every $g\in G$, it follows:

$$\left(\tau g\right).\left(\alpha x\right)=\alpha y\iff g.x=y.$$

Two-graphs are related to graphs in several ways. First, there are graphs associated with two-graphs, and second, there are graphs called descendants of two-graphs. From the graph $\Gamma =(\mathcal{V},\mathcal{E})$, we can construct two-graph $\Phi =(\mathcal{V},\Delta )$ by defining a triple of $\mathcal{V}$ as coherent if the triple induces a subgraph in $\Gamma$ with an odd number of edges. In this case, we say that two-graph $\Phi =(\mathcal{V},\Delta )$ is associated with two-graph $\Gamma =(\mathcal{V},\mathcal{E})$. If $\Gamma$ is SRG $(v,k,\lambda ,\mu )$, then the associated two-graph is regular if and only if $v=2(2k-\lambda -\mu )$. From a two-graph $\Phi =(\mathcal{V},\Delta )$, we can construct a graph $\Gamma$, called a descendant of $\Phi$, in the following way: we fix $x\in \mathcal{V}$ and let $\mathcal{V}\backslash \left\{x\right\}$ be the vertex set of $\Gamma$, then set for any two other vertices $y,z$ to be adjacent in $\Gamma$ if $\{z,x,y\}$ is coherent in $\Phi$. The two-graph is regular if and only if each of its descendants is SRG $(v-1,k,\lambda ,k/2)$. Conference two-graphs are those that have conference graphs as descendants.

In this paper, we classified SRGs $(45,22,10,11)$, SRGs $(49,24,11,12)$, and SRG $(50,21,8,9)$ that have an abelian group of order six as the automorphism group. We also enumerated regular two-graphs on 45 vertices with at least one descendant with automorphisms of order six and regular two-graphs on 50 vertices that have at least one descendant with automorphisms of order six or at least one strongly regular graph with automorphisms of order six associated with it. Since SRGs $(45,22,10,11)$ and SRGs $(49,24,11,12)$ are conference graphs, regular two-graphs on 46 and on 50 vertices are also conference two-graphs.

2. Materials and Methods

Here, we will give a brief description of the methods we used in this work: the construction of strongly regular graphs having an automorphism group of composite order, from their orbit matrices, then the construction of two-graphs from strongly regular graphs and the construction of descendants of two-graphs.

2.1. Construction of Strongly Regular Graphs Admitting an Automorphism Group of Composite Order

Here, we give a brief review of the method taken from [8].

For the construction of strongly regular graphs, we used the method presented in [8]: construction of strongly regular graphs, having an automorphism group of composite order, from their orbit matrices. This is a generalization of the method presented by Behbahani and Lam in [15,16] for constructing strongly regular graphs with prime-order automorphisms from their orbit matrices. For our construction, we used the definition of column orbit matrices introduced in [17].

Definition 1. 

A $(b\times b)$-matrix $C=\left[c_{ij}\right]$ with entries satisfying the conditions:

$$\begin{array}{cc}\hfill \sum _{i=1}^b{c}_{ij}& =\sum _{j=1}^b\frac{n_j}{n_i}{c}_{ij}=k\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \sum _{s=1}^b\frac{n_s}{n_j}{c}_{is}{c}_{js}& ={\delta }_{ij}(k-\mu )+\mu n_i+(\lambda -\mu )c_{ij}\hfill \end{array}$$

(2)

where $0\le c_{ij}\le n_i$, $0\le c_{ii}\le n_i-1$, and ${\sum }_{i=1}^b{n}_i=v$ is called a column orbit matrix for a strongly regular graph with parameters $(v,k,\lambda ,\mu )$ and the orbit length distribution $(n_1,\dots ,n_b)$.

We give here a brief overview of the steps to construct strongly regular graphs with an abelian group of order six as the automorphism group [8]. Such an automorphism acts in orbits of at most four different lengths: 1,2, 3, and 6. If for the action of the group $Z_6$, there are $d_1$ orbits of length 1, $d_2$ orbits of length 2, $d_3$ orbits of length 3, and $d_6$ orbits of length 6, then we denote the corresponding orbit length distribution by $(d_1,d_2,d_3,d_6)$.

Having assumed that $Z_6$ is the automorphism group of SRG $(v,k,\lambda ,\mu )$, we checked all possible orbit length distributions $(d_1,d_2,d_3,d_6)$ for the action of $Z_6$ on SRG $(v,k,\lambda ,\mu )$. In determining the possible orbit length distribution, we considered that 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 ([15], Theorem 3.7). Thus, we need to solve the system of linear equations:

$$\begin{array}{c}{d}_1+2d_2+3d_3+6d_6=v\\ d_1\le \frac{max(\lambda ,\mu )}{k-r}v\\ d_1+2d_3\le \frac{max(\lambda ,\mu )}{k-r}v\\ d_1+3d_3\le \frac{max(\lambda ,\mu )}{k-r}v\end{array}$$

(3)

Next, we need to find prototypes. A prototype for a row of a column orbit matrix C gives us information about the number of occurrences of each integer as an entry of a particular row of C [15].

In [8], we introduced the concept of a prototype for a row of a column orbit matrix $C=\left[c_{ij}\right]$ of a strongly regular graph with a presumed composite order automorphism group.

According to [8]: A prototype of a fixed row (a row corresponding to an orbit of length one) for the distribution $(d_1,d_2,d_3,d_6)$ is a vector

$$p_1=(x_0,x_1,y_0^{\left(2\right)},y_1^{\left(2\right)},y_0^{\left(3\right)},y_1^{\left(3\right)},y_0^{\left(4\right)},y_1^{\left(4\right)})$$

whose components are nonnegative integer solutions of linear equations:

$$\begin{array}{c}{x}_0+x_1=d_1,\\ y_0^{\left(i\right)}+y_1^{\left(i\right)}=d_i, i=2,3,6,\\ x_1+2·y_1^{\left(2\right)}+3·y_1^{\left(3\right)}+6·y_1^{\left(6\right)}=k.\end{array}$$

(4)

where $x_0$ and $x_1$ are the number of zeros and ones, respectively, in the fixed columns of a fixed row and $y_0^{\left(i\right)},y_1^{\left(i\right)}$ are the number of zeros and ones, respectively, in the columns corresponding to orbits of length $i\in \{2,3,6\}$ of a fixed row [8].

According to [8]: A prototype of a row r corresponding to an orbit of length $n_r$, for the distribution $(d_1,d_2,d_3,d_6)$ is a vector

$$p_{n_r}=(x_0,x_{n_r},y_0^{\left(2\right)},\dots ,y_{n_r}^{\left(2\right)},y_0^{\left(3\right)},\dots ,y_{n_r}^{\left(3\right)},y_0^{\left(6\right)},\dots ,y_{n_r}^{\left(6\right)}),$$

satisfying this system of linear equations:

$$\begin{array}{c}{x}_0+x_{n_r}=d_1.\\ {\displaystyle \sum _{e=0}^{n_r}}{y}_e^{\left(i\right)}=d_i, i=2,3,6.\\ x_{n_r}+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(2\right)}·h·\frac{2}{n_r}+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(3\right)}·h·\frac{3}{n_r}+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(6\right)}·h·\frac{6}{n_r}=k,\\ n_r^2{x}_{n_r}+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(2\right)}·h^2·2+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(3\right)}·h^2·3+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(6\right)}·h^2·6=(k-\mu )n_r+\mu n_r^2+(\lambda -\mu )c_{rr}{n}_r\end{array}$$

(5)

for any nonfixed row r, where $x_0$ and $x_{n_r}$ are the number of zeros and $n_r$, respectively, in the fixed columns of row r, and $y_i^{\left(h\right)},i=0,1,\dots ,n_r,$ are the number of is in the columns corresponding to the orbits of length h of row r. Thus, for different $c_{rr}$, we obtain different equations.

We found prototypes for each orbit length distribution using Mathematica [18] and eliminated distributions that have no prototypes. In the next step, we constructed the orbit matrices using the prototypes. During this construction, we eliminated mutually G-isomorphic orbit matrices. For the elimination of orbit matrices leading to G-isomorphic SRGs, we used the same method as for the elimination of orbit matrices from G-isomorphic designs (see [19]).

After constructing the orbit matrices, we refined them using the composition series $\left\{1\right\}⊴Z_3⊴Z_6$. First, each orbit of length two is decomposed into two orbits of length one, and each orbit of length six is decomposed into two orbits of length three. Thus, we obtained orbit matrices for the action of the subgroup $Z_3⊲Z_6$. In the last step of the construction, we obtained adjacency matrices of SRG ($v,k,\lambda ,\mu$), which have $Z_6$ as the automorphism group. Finally, we checked the isomorphisms and studied the full automorphism groups of the constructed graphs. For the construction of strongly regular graphs, we used our programs written in GAP [10].

2.2. Regular Two-Graphs and SRGs

In this section, we give a brief description of the construction of two-graphs from graphs related to it (see [11]).

First, we look at the construction from graphs associated with it. From the simple graph $\Gamma =(\mathcal{V},\mathcal{E})$, we can construct two-graph $\Phi =(\mathcal{V},\Delta )$ by defining a triple of $\mathcal{V}$ as coherent if the triple induces a subgraph in $\Gamma$ with an odd number of edges. If $\Gamma$ is SRG $(v,k,\lambda ,\mu )$ such that $v=2(2k-\lambda -\mu )$, then the associated two-graph on v vertices is regular.

Next, we look at the construction of descendants from regular two-graphs and, conversely, the construction of regular two-graphs from their descendants. From a two-graph $\Phi =(\mathcal{V},\Delta )$, we can construct a graph $\Gamma$, called a descendant of $\Phi$ in the following way: we fix $x\in \mathcal{V}$ and let $\mathcal{V}\backslash \left\{x\right\}$ be the vertex set of $\Gamma$, then set for any two other vertices $y,z$ to be adjacent in $\Gamma$ if $\{z,x,y\}$ is coherent in $\Phi$. The two-graph is regular if and only if each of its descendants is SRG $(v-1,k,\lambda ,k/2)$. The construction is reversible: let G be an SRG $(v-1,k,\lambda ,k/2)$, and adjoin a new element x to the set of vertices of G while keeping the same edge set, then apply the above construction—the constructed two-graph on v vertices will be regular.

3. Results and Discussion

In this section, we present the classification of SRGs $(45,22,10,11)$, SRGs $(49,24,11,12)$, and SRGs $(50,21,8,9)$ that have $Z_6$ as the automorphism group by using the method described in Section 2.1. We also enumerated all SRGs $(45,22,10,11)$, SRGs $(49,24,11,12)$, and SRGs $(50,21,8,9)$ with an automorphism of order six and constructed regular two-graphs on 46 and 50 vertices from them. In this way, we found 236 new regular two-graphs on 46 vertices and 51 new regular two-graphs on 50 vertices, and we obtained 3172 new SRG $(45,22,10,11)$ and 398 new SRG $(49,24,11,12)$ as descendants of the corresponding regular two-graphs.

3.1. SRG (45,22,10,11) with $Z_6$ as Automorphism Group

There are 2104 strongly regular graphs with parameters $(45,22,10,11)$ that are descendants of 97 known regular two-graphs with 46 vertices (see [2,20,21]). Moreover, there are 288 SRGs $(45,22,10,11)$ with the symmetric group $S_3$ as the automorphism group (see [9], Theorem 6), and 208 of them are not descendants of known regular two-graphs. As far as we know, these are the only known SRGs $(45,22,10,11)$. The analysis of the known SRGs $(45,22,10,11)$ is presented in

Table 2. Automorphism group of known SRGs $(45,22,10,11)$.

Aut($\Gamma$)	|Aut($\Gamma$)|	#SRGs
E	1	1478
$Z_2$	2	420
$Z_3$	3	160
$Z_4$	4	4
$K_4$	4	22
$S_3$	6	288
$Z_{10}$	10	8
$Z_{12}$	12	4

. The eigenvalues of SRG $(45,22,10,11)$ are $-3.854<2.854<22$.

Here, we present the results of SRGs $(45,22,10,11)$ having $Z_6$ as the automorphism group.

We constructed them using the method described above. First, we determined all permissible orbit length distributions $(d_1,d_2,d_3,d_6)$ by solving the system of equations:

$$\begin{array}{c}{d}_1+2d_2+3d_3+6d_6=45\\ d_1\le \frac{11}{22-2.854}·45\\ d_1+2d_3\le \frac{11}{22-2.854}·45\\ d_1+3d_3\le \frac{11}{22-2.854}·45\end{array}$$

(6)

We obtained 190 possibilities for the distributions and then found the corresponding prototypes for each orbit distribution $(d_1,d_2,d_3,d_6)$ using Mathematica.

A prototype of a fixed row for the distribution $(d_1,d_2,d_3,d_6)$ is a vector

$$p_1=(x_0,x_1,y_0^{\left(2\right)},y_1^{\left(2\right)},y_0^{\left(3\right)},y_1^{\left(3\right)},y_0^{\left(4\right)},y_1^{\left(4\right)})$$

whose components are nonnegative integer solutions of linear equations:

$$\begin{array}{c}{x}_0+x_1=d_1,\\ y_0^{\left(i\right)}+y_1^{\left(i\right)}=d_i, i=2,3,6,\\ x_1+2·y_1^{\left(2\right)}+3·y_1^{\left(3\right)}+6·y_1^{\left(6\right)}=22.\end{array}$$

(7)

A prototype of a row r corresponding to an orbit of length $n_r\in \{2,3,6\}$ for the distribution $(d_1,d_2,d_3,d_6)$ is a vector

$$p_{n_r}=(x_0,x_{n_r},y_0^{\left(2\right)},\dots ,y_{n_r}^{\left(2\right)},y_0^{\left(3\right)},\dots ,y_{n_r}^{\left(3\right)},y_0^{\left(6\right)},\dots ,y_{n_r}^{\left(6\right)}),$$

satisfying this system of linear equations:

$$\begin{array}{c}{x}_0+x_{n_r}=d_1.\\ {\displaystyle \sum _{e=0}^{n_r}}{y}_e^{\left(i\right)}=d_i, i=2,3,6.\\ x_{n_r}+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(2\right)}·h·\frac{2}{n_r}+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(3\right)}·h·\frac{3}{n_r}+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(6\right)}·h·\frac{6}{n_r}=22,\\ n_r^2{x}_{n_r}+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(2\right)}·h^2·2+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(3\right)}·h^2·3+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(6\right)}·h^2·6=11n_r+11n_r^2-c_{rr}{n}_r, c_{rr}\in \{0,\dots ,n_r-1\}.\end{array}$$

(8)

We constructed the orbit matrices row-by-row using the prototypes while eliminating mutually G-isomorphic orbit matrices. In this case, only one distribution leads to any orbit matrices. The next steps are to refine the obtained orbit matrices, i.e., to construct the corresponding orbit matrices for $Z_3$ and, finally, to construct the SRGs with parameters $(45,22,10,11)$. For the construction, we used our programs written in GAP. For each orbit length distribution, we present the number of nonisomorphic orbit matrices for $Z_6$, the number of orbit matrices for $Z_3$, and the number of constructed SRGs with parameters $(45,22,10,11)$ in

Table 3. Orbit matrices of SRGs $(45,22,10,11)$ having automorphism group $Z_6$.

Distribution	#OM-${\mathit{Z}}_6$	#OM-${\mathit{Z}}_3$	#SRGs
(1,4,4,4)	7	7	208

.

Using GAP, we checked isomorphisms of strongly regular graphs and compared them with known SRG $(45,22,10,11)$, then found that, among the constructed strongly regular graphs, there are 208 mutually nonisomorphic graphs, 204 of which are new. The results of our construction and an analysis of the full automorphism groups of the 208 constructed graphs are summarized in Theorem 1.

Theorem 1. 

Up to isomorphism, there are exactly 208 strongly regular graphs with parameters (45, 22, 10, 11) whose automorphism group is isomorphic to a cyclic group of order six. The full automorphism group of these graphs is presented in

Table 4. SRGs with parameters $(45,22,10,11)$ having $Z_6$ as the automorphism group.

Aut($\Gamma$)	|Aut($\Gamma$)|	#SRGs
$Z_6$	6	176
$Z_{12}$	12	32

.

The adjacency matrices of the constructed SRGs are available online (accessed on 25 January 2022): http://www.math.uniri.hr/~mmaksimovic/45_z6.txt.

We analyzed the SRGs $(45,22,10,11)$ having $Z_6$ or $S_3$ as the automorphism group and eliminated the isomorphic SRGs using our programs written in GAP. The results are summarized in the following theorem.

Theorem 2. 

Up to isomorphism, there are exactly 496 strongly regular graphs with parameters (45,22,10,11) whose automorphism group has order six.

3.2. SRG (49, 24, 11, 12) with $Z_6$ as Automorphism Group

According to [22], there are 54 regular two-graphs on 50 vertices giving 785 descendants, i.e., SRGs $(49,24,11,12)$. In addition, there are 72 SRGs $(49,24,11,12)$ having the symmetric group of order six as the automorphism group (see [9], Theorem 8), and 50 of them are not descendants of known regular two-graphs. As far as we know, these are the only known SRGs $(49,24,11,12)$. The analysis of the known SRGs $(49,24,11,12)$ is presented in

Table 5. Automorphism group of known SRGs $(49,24,11,12)$.

Aut($\Gamma$)	|Aut($\Gamma$)|	#SRGs	Aut($\Gamma$)	|Aut($\Gamma$)|	#SRGs
E	1	592	$S_4$	24	4
$Z_2$	2	24	$SL(2,3)$	24	6
$Z_3$	3	106	$Z_3\times D_{16}$	48	2
$Z_6$	6	12	$Z_3\times (Z_7:Z_3)$	63	2
$S_3$	6	42	$Z_3\times SL(2,3)$	72	1
$Z_3\times Z_3$	9	14	$S_3\times (Z_7:Z_3)$	126	4
$Z_3\times S_3$	18	22	$(Z_7\times Z_7):(Z_3\times D_{16})$	2352	1
$Z_7:Z_3$	21	2	$(Z_7\times Z_7):(Z_3\times SL(2,3))$	3528	1

. The eigenvalues of SRG $(49,24,11,12)$ are $-4<3<24$.

Here, we present all SRGs $(49,24,11,12)$ with $Z_6$ as the automorphism group and show that there are exactly 99 of them. We constructed them using the method described above. First, we checked all permissible orbit length distributions $(d_1,d_2,d_3,d_6)$ by solving the system of equations:

$$\begin{array}{c}{d}_1+2d_2+3d_3+6d_6=49\\ d_1\le \frac{12}{24-3}·49\\ d_1+2d_3\le \frac{12}{24-3}·49\\ d_1+3d_3\le \frac{12}{24-3}·49\end{array}$$

(9)

We obtained 259 possibilities for distributions and then found the corresponding prototypes for each orbit distribution $(d_1,d_2,d_3,d_6)$ using Mathematica.

A prototype of a fixed row for the distribution $(d_1,d_2,d_3,d_6)$ is a vector

$$p_1=(x_0,x_1,y_0^{\left(2\right)},y_1^{\left(2\right)},y_0^{\left(3\right)},y_1^{\left(3\right)},y_0^{\left(4\right)},y_1^{\left(4\right)})$$

whose components are nonnegative integer solutions of linear equations:

$$\begin{array}{c}{x}_0+x_1=d_1,\\ y_0^{\left(i\right)}+y_1^{\left(i\right)}=d_i, i=2,3,6,\\ x_1+2·y_1^{\left(2\right)}+3·y_1^{\left(3\right)}+6·y_1^{\left(6\right)}=24.\end{array}$$

(10)

A prototype of a row r corresponding to an orbit of length $n_r\in \{2,3,6\}$ for the distribution $(d_1,d_2,d_3,d_6)$ is a vector

$$p_{n_r}=(x_0,x_{n_r},y_0^{\left(2\right)},\dots ,y_{n_r}^{\left(2\right)},y_0^{\left(3\right)},\dots ,y_{n_r}^{\left(3\right)},y_0^{\left(6\right)},\dots ,y_{n_r}^{\left(6\right)}),$$

satisfying this system of linear equations:

$$\begin{array}{c}{x}_0+x_{n_r}=d_1.\\ {\displaystyle \sum _{e=0}^{n_r}}{y}_e^{\left(i\right)}=d_i, i=2,3,6.\\ x_{n_r}+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(2\right)}·h·\frac{2}{n_r}+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(3\right)}·h·\frac{3}{n_r}+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(6\right)}·h·\frac{6}{n_r}=24,\\ n_r^2{x}_{n_r}+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(2\right)}·h^2·2+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(3\right)}·h^2·3+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(6\right)}·h^2·6=12n_r+12n_r^2-c_{rr}{n}_r, c_{rr}\in \{0,\dots ,n_r-1\}.\end{array}$$

(11)

We constructed the orbit matrices row-by-row using the prototypes while eliminating mutually G-isomorphic orbit matrices. We obtained 259 possibilities for orbit length distributions, but only a few of them led to orbit matrices. The next steps were to refine the obtained orbit matrices, i.e., construct the corresponding orbit matrices for $Z_3$ and, finally, construct the SRGs with parameters $(49,24,11,12)$. For the construction, we used our programs written in GAP. For each orbit length distribution, we present the number of nonisomorphic orbit matrices for $Z_6$, the number of orbit matrices for $Z_3$, and the number of constructed SRGs with parameters $(49,24,11,12)$ in

Table 6. Orbit matrices of SRGs $(49,24,11,12)$ having automorphism group $Z_6$.

Distr.	#OM-${\mathit{Z}}_6$	#OM-${\mathit{Z}}_3$	#SRGs	Distr.	#OM-${\mathit{Z}}_6$	#OM-${\mathit{Z}}_3$	#SRGs
(0, 2, 3, 6)	8	16	0	(1, 3, 6, 4)	2	2	0
(0, 2, 5, 5)	4	0	0	(1, 6, 0, 6)	1	0	0
(0, 2, 7, 4)	8	0	0	(3, 2, 0, 7)	4	10	0
(1, 0, 0, 8)	2	15	59	(3, 2, 6, 4)	6	16	0
(1, 0, 2, 7)	20	32	37	(5, 1, 0, 7)	2	4	0
(1, 0, 8, 4)	26	24	0	(5, 1, 6, 4)	2	2	0
(1, 3, 0, 7)	6	6	10	(7, 0, 0, 7)	2	2	0
(1, 3, 2, 6)	10	2	0

.

Using GAP, we checked the isomorphisms of strongly regular graphs and compared them with known SRG $(49,24,11,12)$, then found that, among the constructed strongly regular graphs, there are 99 mutually nonisomorphic graphs, 50 of which are new. The results of our construction and an analysis of the full automorphism groups of the constructed 99 graphs are summarized in the following statement.

Theorem 3. 

Up to isomorphism, there are exactly 99 strongly regular graphs with parameters (49,24,11,12) whose automorphism group is isomorphic to a cyclic group of order six. The full automorphism group of these graphs is presented in

Table 7. SRGs with parameters $(49,24,11,12)$ having $Z_6$ as the automorphism group.

Aut($\Gamma$)	|Aut($\Gamma$)|	#SRGs
$Z_6$	6	60
$Z_6\times Z_2$	12	2
$Z_3\times S_3$	18	22
$SL(2,3)$	24	6
$Z_3\times D_{16}$	48	2
$Z_3\times SL(2,3)$	72	1
$S_3\times (Z_7:Z_3)$	126	4
$(Z_7\times Z_7):(Z_3\times D_{16})$	2352	1
$(Z_7\times Z_7):(Z_3\times SL(2,3))$	3528	1

.

The adjacency matrices of the constructed SRGs are available online (accessed on 25 January 2022): http://www.math.uniri.hr/~mmaksimovic/49_z6.txt.

We analyzed the SRGs $(49,24,11,12)$ having $Z_6$ or $S_3$ as the automorphism group and eliminated the isomorphic SRGs using our programs written in GAP. The results are summarized in Theorem 4.

Theorem 4. 

Up to isomorphism, there are exactly 145 strongly regular graphs with parameters (49,24,11,12) having an automorphism group of order six.

3.3. SRG (50,21,8,9) with $Z_6$ as Automorphism Group

There are exactly 18 Steiner $(2,4,25)$ systems, and 18 SRGs ($50,21,8,9$) obtained from them can be found in [22], while 3 of them have a nonabelian automorphism group of order six. Moreover, there are 42 other SRGs ($50,21,8,9$) with a nonabelian automorphism group of order six (see [9], Theorem 9). In

Table 8. Automorphism group of known SRGs $(50,21,8,9)$.

Aut($\Gamma$)	|Aut($\Gamma$)|	#SRGs	Aut($\Gamma$)	|Aut($\Gamma$)|	#SRGs
E	1	2	$Z_7:Z_3$	21	1
$Z_3$	3	8	$Z_3\times (Z_7:Z_3)$	63	1
$S_3$	6	35	$Z_3\times S_4$	72	1
$Z_3\times Z_3$	9	3	$(Z_5\times Z_5):S_3$	150	1
$(Z_3\times Z_3):Z_2$	18	3	$PSL(3,2):Z_2$	336	1
$Z_3\times S_3$	18	3	$Z_3\times PSL(3,2)$	504	1

, we present the analysis of the known SRGs $(50,21,8,9)$. The eigenvalues of SRG $(50,21,8,9)$ are $-4<3<21$.

Here, we present the 51 constructed SRGs $(50,21,8,9)$ that have an abelian automorphism group of order six.

We constructed them using the method described above. First, we checked all permissible orbit length distributions $(d_1,d_2,d_3,d_6)$ by solving the system of equations:

$$\begin{array}{c}{d}_1+2d_2+3d_3+6d_6=50\\ d_1\le \frac{9}{21-3}·50\\ d_1+2d_3\le \frac{9}{21-3}·50\\ d_1+3d_3\le \frac{9}{21-3}·50\end{array}$$

(12)

We obtained 170 possibilities for the distributions and then found the corresponding prototypes for each orbit distribution $(d_1,d_2,d_3,d_6)$ using Mathematica.

A prototype of a fixed row for the distribution $(d_1,d_2,d_3,d_6)$ is a vector

$$p_1=(x_0,x_1,y_0^{\left(2\right)},y_1^{\left(2\right)},y_0^{\left(3\right)},y_1^{\left(3\right)},y_0^{\left(4\right)},y_1^{\left(4\right)})$$

whose components are nonnegative integer solutions of linear equations:

$$\begin{array}{c}{x}_0+x_1=d_1,\\ y_0^{\left(i\right)}+y_1^{\left(i\right)}=d_i, i=2,3,6,\\ x_1+2·y_1^{\left(2\right)}+3·y_1^{\left(3\right)}+6·y_1^{\left(6\right)}=21.\end{array}$$

(13)

A prototype of a row r corresponding to an orbit of length $n_r\in \{2,3,6\}$ for the distribution $(d_1,d_2,d_3,d_6)$ is a vector

$$p_{n_r}=(x_0,x_{n_r},y_0^{\left(2\right)},\dots ,y_{n_r}^{\left(2\right)},y_0^{\left(3\right)},\dots ,y_{n_r}^{\left(3\right)},y_0^{\left(6\right)},\dots ,y_{n_r}^{\left(6\right)}),$$

satisfying this system of linear equations:

$$\begin{array}{c}{x}_0+x_{n_r}=d_1.\\ {\displaystyle \sum _{e=0}^{n_r}}{y}_e^{\left(i\right)}=d_i, i=2,3,6.\\ x_{n_r}+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(2\right)}·h·\frac{2}{n_r}+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(3\right)}·h·\frac{3}{n_r}+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(6\right)}·h·\frac{6}{n_r}=21,\\ n_r^2{x}_{n_r}+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(2\right)}·h^2·2+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(3\right)}·h^2·3+{\displaystyle \sum _{h=1}^{n_r}}{y}_h^{\left(6\right)}·h^2·6=12n_r+9n_r^2-c_{rr}{n}_r, c_{rr}\in \{0,\dots ,n_r-1\}.\end{array}$$

(14)

We constructed the orbit matrices row-by-row using the prototypes while eliminating mutually G-isomorphic orbit matrices. Only a few of the orbit length distributions led to orbit matrices. The next steps were to refine the obtained orbit matrices, i.e., construct the corresponding orbit matrices for $Z_3$ and, finally, construct the SRGs with parameters $(50,21,8,9)$. For the construction, we used our programs written in GAP. For each orbit length distribution, we present the number of nonisomorphic orbit matrices for $Z_6$, the number of orbit matrices for $Z_3$, and the number of constructed SRGs with parameters $(50,21,8,9)$ in

Table 9. Orbit matrices of SRGs $(50,21,8,9)$ having automorphism group $Z_6$.

Distr.	#OM-${\mathit{Z}}_6$	#OM-${\mathit{Z}}_3$	#SRGs	Distr.	#OM-${\mathit{Z}}_6$	#OM-${\mathit{Z}}_3$	#SRGs
(0, 1, 2, 7)	10	3	0	(1, 2, 5, 5)	2	0	0
(0, 1, 4, 6)	10	4	10	(1, 2, 7, 4)	4	0	0
(0, 1, 6, 5)	12	21	0	(2, 0, 2, 7)	10	16	19
(0, 1, 8, 4)	8	8	0	(2, 3, 0, 7)	2	3	12
(0, 4, 2, 6)	2	2	10	(2, 3, 2, 6)	6	1	0
(0, 4, 4, 5)	4	3	0	(2, 3, 6, 4)	2	4	0
(0, 4, 6, 4)	3	4	0	(4, 2, 6, 4)	6	12	0
(0, 4, 8, 3)	4	6	0	(6, 1, 6, 4)	1	1	0
(1, 2, 3, 6)	10	20	0

.

Using GAP, we checked the isomorphisms of strongly regular graphs and compared them with known SRG $(50,21,8,9)$, then found that, among the constructed strongly regular graphs, there are 51 mutually nonisomorphic graphs, 45 of which are new. The results of our construction and an analysis of the full automorphism groups of the 51 constructed graphs are summarized in Theorem 5.

Theorem 5. 

Up to isomorphism, there are exactly 51 strongly regular graphs with parameters (50, 21, 8, 9) whose automorphism group is isomorphic to a cyclic group of order six. The full automorphism group of these graphs is presented in

Table 10. SRGs with parameters $(50,21,8,9)$ having $Z_6$ as the automorphism group.

Aut($\Gamma$)	|Aut($\Gamma$)|	#SRGs
$Z_6$	6	38
$Z_{12}$	12	5
$Z_3\times S_3$	18	3
$Z_3\times A_4$	36	1
$Z_7:Z_6$	42	1
$Z_3\times S_4$	72	1
$PSL(3,2):Z_2$	336	1
$Z_3\times PSL(3,2)$	504	1

.

The adjacency matrices of the constructed SRGs are available online (accessed on 25 January 2022): http://www.math.uniri.hr/~mmaksimovic/50_z6.txt.

By analyzing the SRGs $(50,21,8,9)$ having $Z_6$ or $S_3$ as the automorphism group and eliminating the isomorphic SRGs with GAP, we obtained the results summarized in Theorem 6.

Theorem 6. 

Up to isomorphism, there are exactly 90 strongly regular graphs with parameters (50, 21, 8, 9) whose automorphism group is of order six.

3.4. Regular Two-Graphs on 46 Vertices

There are at least 97 regular two-graphs on 46 vertices (see [2,20,21]), yielding 2104 descendants that are strongly regular graphs with parameters $(45,22,10,11)$. Representatives of the descendants can be found in [2,21,23]. The analysis of the known regular two-graphs on 46 vertices is shown in

Table 11. Known regular two-graphs on 46 vertices.

$\#\Phi$	$\left|\mathbf{Aut}\right(\Phi \left)\right|$	Descendants of $\Phi$	“S”
56	2	$[20\times E,6\times Z_2]$	NO
16	3	$[12\times E,10\times Z_3]$	NO
8	10	$[4\times E,1\times Z_2,1\times Z_{10}]$	NO
5	2	$[22\times E,2\times Z_2]$	YES
2	4	$[6\times E,11\times Z_2]$	YES
4	8	$[2\times E,6\times Z_2,2\times K_4,1\times Z_4]$	YES
2	8	$[2\times E,6\times Z_2,3\times K_4]$	YES
4	24	$[2\times Z_2,2\times K_4,2\times S_3,1\times Z_{12}]$	YES

. The second column of the table contains the order of the corresponding full automorphism group of the regular two-graph $\Phi$, while in the third column, we give the number of non-isomorphic descendants of $\Phi$ with a given full automorphism group. In the last column, we indicate whether a $\Phi$ is self-complementary or not, and in the first column, we give the number of regular two-graphs $\Phi$ that have the given properties.

In Section 3.4.1, we analyze SRGs $(45,22,10,11)$ with $S_3$ or $Z_6$ as the automorphism group and construct the corresponding two-graphs. In this way, we classified regular two-graphs on 46 vertices that have at least one descendant with an automorphism group of order six, while increasing the number of known regular two-graphs on 46 vertices from 97 to 333 and the number of known SRGs $(45,22,10,11)$ from 2104 to 5276.

3.4.1. Descendants with an Automorphism of Order Six

From Theorem 2, we know that there are 496 strongly regular graphs with parameters $(45,22,10,11)$ with an automorphism group of order six. We analyzed these strongly regular graphs and obtained the following result.

Theorem 7. 

Up to isomorphism, there are exactly 240 regular two-graphs on 46 vertices that have at least one descendant with an automorphism group of order six, and among them, there are 14 self-complementary regular two-graphs. They give rise to 3200 strongly regular graphs with parameters (45, 22, 10, 11).

Proof of Theorem 7. 

From 496 SRGs $(45,22,10,11)$ with an automorphism group of order six, we constructed the corresponding regular two-graphs on 45 vertices and eliminated the isomorphic ones. We obtained 240 regular two-graphs ${\Phi }_i,i\in \{1,2,\dots ,240\}$, and among them, there are 14 self-complementary two-graphs and 113 pairs of complementary two-graphs. We constructed 3200 descendants from these regular two-graphs. The results on the constructed regular two-graphs are summarized in Table 12, and in the table, each pair of complementary two-graphs is represented by one of them.

For each two-graph from Table 12, the adjacency matrix of one of its descendants is available online (accessed on 25 January 2022): http://www.math.uniri.hr/~mmaksimovic/46_descendants6.txt. □

Table 12. Descendants of the regular two-graphs on 46 vertices.

i	$\left|\mathbf{Aut}\right({\Phi }_{\mathit{i}}\left)\right|$	Descendants of ${\Phi }_{\mathit{i}}$	“S”
1–42	6	$[4\times E,4\times Z_2,4\times Z_3,2\times Z_6]$	NO
43–102	6	$[4\times E,4\times Z_2,4\times Z_3,2\times S_3]$	NO
103–106	12	$[1\times E,3\times Z_2,1\times Z_3,2\times K_4,1\times Z_6,2\times S_3]$	NO
107–111	12	$[2\times E,2\times Z_2,2\times Z_3,2\times Z_{12}]$	NO
112–113	24	$[2\times Z_2,2\times K_4,2\times S_3,1\times Z_{12}]$	NO
114–121	6	$[4\times E,4\times Z_2,4\times Z_3,2\times S_3]$	YES
122–123	12	$[2\times E,2\times Z_2,2\times Z_3,2\times Z_{12}]$	YES
124–127	24	$[2\times Z_2,2\times K_4,2\times S_3,1\times Z_{12}]$	YES

Table 12. Descendants of the regular two-graphs on 46 vertices.

i	$\left|\mathbf{Aut}\right({\Phi }_{\mathit{i}}\left)\right|$	Descendants of ${\Phi }_{\mathit{i}}$	“S”
1–42	6	$[4\times E,4\times Z_2,4\times Z_3,2\times Z_6]$	NO
43–102	6	$[4\times E,4\times Z_2,4\times Z_3,2\times S_3]$	NO
103–106	12	$[1\times E,3\times Z_2,1\times Z_3,2\times K_4,1\times Z_6,2\times S_3]$	NO
107–111	12	$[2\times E,2\times Z_2,2\times Z_3,2\times Z_{12}]$	NO
112–113	24	$[2\times Z_2,2\times K_4,2\times S_3,1\times Z_{12}]$	NO
114–121	6	$[4\times E,4\times Z_2,4\times Z_3,2\times S_3]$	YES
122–123	12	$[2\times E,2\times Z_2,2\times Z_3,2\times Z_{12}]$	YES
124–127	24	$[2\times Z_2,2\times K_4,2\times S_3,1\times Z_{12}]$	YES

Using our programs written in GAP, we compared the constructed two-graph with already known regular two-graphs on 46 vertices and found that the graphs ${\Phi }_{124},{\Phi }_{125},{\Phi }_{126}$, and ${\Phi }_{127}$ were already known, so 236 of the constructed two-graphs are new. The results obtained are summarized in the following theorem.

Theorem 8. 

Up to isomorphism, there are at least 333 regular two-graphs on 46 vertices. Among them there are 27 self-complementary two-graphs, and they give rise to 5276 nonisomorphic descendants.

3.5. Regular Two-Graphs on 50 Vertices

There are at least 54 regular two-graphs on 50 vertices yielding 785 descendants that are strongly regular graphs with parameters $(49,24,11,12)$. Representatives of the descendants can be found in [23]. The analysis of known regular two-graphs on 50 vertices is shown in

Table 13. Known regular two-graphs on 50 vertices.

$\#\Phi$	$\left|\mathbf{Aut}\right(\Phi \left)\right|$	Descendants of $\Phi$	“S”
2	1	$[50\times E]$	NO
1	3	$[14\times E,8\times Z_3]$	NO
2	3	$[15\times E,5\times Z_3]$	NO
6	3	$[16\times E,2\times Z_3]$	NO
1	6	$[4\times E,8\times Z_2,2\times S_3]$	NO
1	6	$[7\times E,3\times Z_3,2\times Z_6]$	NO
1	6	$[7\times E,2\times Z_2,2\times Z_6]$	NO
1	6	$[8\times E,2\times Z_6]$	NO
3	9	$[4\times E,4\times Z_3,2\times (Z_3\times Z_3)]$	NO
1	18	$[2\times E,1\times Z_3,2\times (Z_3\times S_3),2\times S_3]$	NO
1	21	$[1\times E,4\times Z_3,1\times (Z_7:Z_3)]$	NO
1	24	$[2\times E,2\times SL(2,3\left)\right]$	NO
1	63	$[2\times Z_3,1\times (Z_3\times (Z_7:Z_3)),1\times (Z_3\times Z_3)]$	NO
1	126	$[1\times Z_3,1\times (Z_3\times S_3),1\times (S_3\times (Z_7:Z_3))]$	NO
1	1008	$[1\times (S_3\times (Z_7:Z_3)),1\times S_4]$	NO
1	2	$[24\times E,2\times Z_2]$	YES
1	24	$[2\times E,2\times SL(2,3\left)\right]$	YES
1	48	$[2\times Z_2,2\times (Z_3\times D_{16})]$	YES
1	150	$[2\times S_3]$	YES
1	3528	$[1\times (Z_7\times Z_7):(Z_3\times SL(2,3)),1\times (Z_3\times SL(2,3))]$	YES
1	117,600	$[1\times (Z_7\times Z_7):(Z_3\times D_{16})]$	YES

. The second column of the table contains the order of the corresponding full automorphism group of the regular two-graph $\Phi$, while in the third column, we give the number of non-isomorphic descendants of $\Phi$ with a given full automorphism group. In the last column, we indicate whether a $\Phi$ is self-complementary or not, and in the first column, we give the number of regular two-graphs $\Phi$ having the given properties.

Since it holds for SRG $(50,21,8,9)$ that $50=2(2\times 21-8-9)$, the associated two-graph is a regular two-graph on 50 vertices.

In Section 3.5.1, we analyze SRGs $(49,24,11,12)$ and SRGs $(50,21,8,9)$ with $S_3$ or $Z_6$ as the automorphism group and construct the corresponding two-graphs. In this way, we classified regular two-graphs on 50 vertices that have at least one descendant with an automorphism group of order six or at least one graph associated with it having an automorphism group of order six and increase the number of known regular two-graphs on 50 vertices from 54 to 105 and the number of known SRGs $(49,24,11,12)$ from 835 to 1233.

3.5.1. Descendants with an Automorphism of Order Six

We analyzed 145 SRGs $(49,24,11,12)$ and 90 SRGs $(50,21,8,9)$ with an automorphism of order six and obtained the following result.

Theorem 9. 

Up to isomorphism, there are exactly 72 regular two-graphs on 50 vertices that have at least one descendant with an automorphism group of order six or at least one graph associated with it having an automorphism group of order six. Among them, there are 10 self-complementary regular two-graphs, and they give rise to 587 strongly regular graphs with parameters (49,24,11,12).

Proof of Theorem 9. 

From SRGs $(49,24,11,12)$ and SRGs $(50,21,8,9)$ with an automorphism group of order six, we constructed the corresponding two-graphs and eliminated the isomorphic ones. We obtained 72 regular two-graphs ${\Phi }_i,i\in \{1,2,\dots ,72\}$, and among them, there are 10 self-complementary two-graphs and 31 pairs of complementary two-graphs. We constructed 587 descendants from these regular two-graphs. The results on the constructed regular two-graphs are summarized in Table 14, and in the table, each pair of complementary two-graphs is represented by one of them.

For each two-graph from Table 14, the adjacency matrix of one of its descendants is available online (accessed on 25 January 2022): http://www.math.uniri.hr/~mmaksimovic/50_descendants6.txt. □

Table 14. Descendants of the regular two-graphs on 50 vertices.

i	$\left|\mathbf{Aut}\right({\Phi }_{\mathit{i}}\left)\right|$	Descendants of ${\Phi }_{\mathit{i}}$	“S”
1–2	6	$[4\times E,8\times Z_2,2\times S_3]$	NO
3–4	6	$[5\times E,6\times Z_2,1\times Z_3]$	NO
5	6	$[6\times E,4\times Z_2,1\times Z_3]$	NO
6–7	6	$[7\times E,2\times Z_2,2\times Z_6]$	NO
8	6	$[7\times E,3\times Z_3,2\times Z_6]$	NO
9–18	6	$[8\times E,2\times Z_6]$	NO
19	12	$[1\times E,5\times Z_2,1\times Z_6,3\times S_3]$	NO
20	18	$[1\times E,2\times Z_2,1\times Z_3,1\times (Z_3\times Z_3),2\times S_3]$	NO
21–23	18	$[2\times E,1\times Z_3,2\times (Z_3\times S_3),2\times S_3]$	NO
24	24	$[2\times E,2\times SL(2,3\left)\right]$	NO
25	24	$[1\times E,1\times Z_2,1\times Z_4,2\times S_3]$	NO
26	24	$[1\times E,2\times Z_2,1\times (Z_2\times Z_6)]$	NO
27	36	$[2\times Z_2,1\times (Z_3\times S_3),2\times S_3]$	NO
28	72	$[1\times A_4,1\times Z_2,2\times (Z_3\times S_3)]$	NO
29	126	$[1\times Z_3,1\times (Z_3\times S_3),1\times (S_3\times (Z_7:Z_3))]$	NO
30	144	$[1\times K_4,1\times (Z_3\times S_3),1\times S_4]$	NO
31	1008	$[1\times (S_3\times (Z_7:Z_3)),1\times S_4]$	NO
32	6	$[4\times E,8\times Z_2,2\times S_3]$	YES
33–34	6	$[8\times E,2\times Z_6]$	YES
35–36	12	$[4\times E,1\times Z_6]$	YES
37	24	$[2\times E,2\times SL(2,3\left)\right]$	YES
38	48	$[2\times Z_2,2\times (Z_3\times D_{16})]$	YES
39	150	$[2\times S_3]$	YES
40	3528	$[1\times (Z_7\times Z_7):(Z_3\times SL(2,3)),1\times (Z_3\times SL(2,3))]$	YES
41	117,600	$[1\times (Z_7\times Z_7):(Z_3\times D_{16})]$	YES

Table 14. Descendants of the regular two-graphs on 50 vertices.

i	$\left|\mathbf{Aut}\right({\Phi }_{\mathit{i}}\left)\right|$	Descendants of ${\Phi }_{\mathit{i}}$	“S”
1–2	6	$[4\times E,8\times Z_2,2\times S_3]$	NO
3–4	6	$[5\times E,6\times Z_2,1\times Z_3]$	NO
5	6	$[6\times E,4\times Z_2,1\times Z_3]$	NO
6–7	6	$[7\times E,2\times Z_2,2\times Z_6]$	NO
8	6	$[7\times E,3\times Z_3,2\times Z_6]$	NO
9–18	6	$[8\times E,2\times Z_6]$	NO
19	12	$[1\times E,5\times Z_2,1\times Z_6,3\times S_3]$	NO
20	18	$[1\times E,2\times Z_2,1\times Z_3,1\times (Z_3\times Z_3),2\times S_3]$	NO
21–23	18	$[2\times E,1\times Z_3,2\times (Z_3\times S_3),2\times S_3]$	NO
24	24	$[2\times E,2\times SL(2,3\left)\right]$	NO
25	24	$[1\times E,1\times Z_2,1\times Z_4,2\times S_3]$	NO
26	24	$[1\times E,2\times Z_2,1\times (Z_2\times Z_6)]$	NO
27	36	$[2\times Z_2,1\times (Z_3\times S_3),2\times S_3]$	NO
28	72	$[1\times A_4,1\times Z_2,2\times (Z_3\times S_3)]$	NO
29	126	$[1\times Z_3,1\times (Z_3\times S_3),1\times (S_3\times (Z_7:Z_3))]$	NO
30	144	$[1\times K_4,1\times (Z_3\times S_3),1\times S_4]$	NO
31	1008	$[1\times (S_3\times (Z_7:Z_3)),1\times S_4]$	NO
32	6	$[4\times E,8\times Z_2,2\times S_3]$	YES
33–34	6	$[8\times E,2\times Z_6]$	YES
35–36	12	$[4\times E,1\times Z_6]$	YES
37	24	$[2\times E,2\times SL(2,3\left)\right]$	YES
38	48	$[2\times Z_2,2\times (Z_3\times D_{16})]$	YES
39	150	$[2\times S_3]$	YES
40	3528	$[1\times (Z_7\times Z_7):(Z_3\times SL(2,3)),1\times (Z_3\times SL(2,3))]$	YES
41	117,600	$[1\times (Z_7\times Z_7):(Z_3\times D_{16})]$	YES

Using our programs written in GAP, we compared the constructed regular two-graphs with known regular two-graphs on 50 vertices and found that 21 graphs:

$${\Phi }_2,\overline{{\Phi }_2},{\Phi }_6,\overline{{\Phi }_6},{\Phi }_8,\overline{{\Phi }_8},{\Phi }_9,\overline{{\Phi }_9},{\Phi }_{21},\overline{{\Phi }_{21}},{\Phi }_{26},\overline{{\Phi }_{26}},{\Phi }_{29},\overline{{\Phi }_{29}},{\Phi }_{31},\overline{{\Phi }_{31}},{\Phi }_{37},{\Phi }_{38},{\Phi }_{39},{\Phi }_{40},{\Phi }_{41}$$

were already known, so of the constructed two-graphs, 51 are new. The results are summarized in the following theorem.

Theorem 10. 

Up to isomorphism, there are at least 105 regular two-graphs on 50 vertices. Among them, there are 11 self-complementary two-graphs, leading to 1233 nonisomorphic descendants.

4. Conclusions

In this work, we constructed SRGs $(45,22,10,11)$, SRGs $(49,24,11,12)$, and SRGs $(50,21,$$8,9)$ having $Z_6$ as the automorphism group and compared them with those having $S_3$ as the automorphism group to enumerate the graphs that have an automorphism group of order six. In

Table 15. SRGs on up to 50 vertices that are related to a regular two-graph and have an automorphism group of order six.

($\mathit{v},\mathit{k},\mathit{\lambda },\mathit{\mu }$)	|Aut($\Gamma$)|	#SRGs
$(45,22,10,11)$	6	464
$(45,22,10,11)$	12	32
$(49,24,11,12)$	6	102
$(49,24,11,12)$	12	2
$(49,24,11,12)$	18	22
$(49,24,11,12)$	24	10
$(49,24,11,12)$	48	2
$(49,24,11,12)$	72	1
$(49,24,11,12)$	126	4
$(49,24,11,12)$	2352	1
$(49,24,11,12)$	3528	1
$(50,21,8,9)$	6	73
$(50,21,8,9)$	12	5
$(50,21,8,9)$	18	6
$(50,21,8,9)$	36	1
$(50,21,8,9)$	42	1
$(50,21,8,9)$	72	1
$(50,21,8,9)$	150	1
$(50,21,8,9)$	336	1
$(50,21,8,9)$	504	1

, we summarize the obtained results on the SRGs on up to 50 vertices that are related to a regular two-graph, i.e., we give a list of them and the orders of their full automorphism groups.

We also constructed 236 new regular two-graphs on 46 vertices and 51 new regular two-graphs on 50 vertices and present the updated

Table 16. Known regular two-graphs on up to 50 vertices.

n	$\mathit{N}\left(\mathit{n}\right)$	n	$\mathit{N}\left(\mathit{n}\right)$
6	$\overline{1}$	30	$\overline{6}$
10	$\overline{1}$	36	$\overline{227}$
14	$\overline{1}$	38	194
16	$\overline{1}$	42	752
18	$\overline{1}$	46	333
26	$\overline{4}$	50	105
28	$\overline{1}$

of known regular two-graphs.

In addition, we obtained 3172 new SRG $(45,22,10,11)$ and 398 new SRG $(49,24,11,12)$ as descendants of the corresponding regular two-graphs. Thus, we significantly increased the number of known SRGs with these parameters and contributed to the classification of SRGs and regular two-graphs on up to 50 vertices.