Abstract. Two signed graphs are called switching isomorphic to each other if one is isomorphic to a switching of the other. The wheel $W_n$ is the join of the cycle $C_n$ and a vertex. For $0 \leq p \leq n$, $\psi_{p}(n)$ is defined to be the number of switching non-isomorphic signed $W_n$ with exactly $p$ negative edges on $C_n$. The number of switching non-isomorphic signed $W_n$ is denoted by $\psi(n)$. In this paper, we compute the values of $\psi_{p}(n)$ for $p=0,1,2,3,4,n-4,n-3,n-2,n-1,n$ and of $\psi(n)$ for $n=4,5,...,10$.

Keywords: signed wheel; switching isomorphism; switching isomorphism.

§ INTRODUCTION

A signed graph, denoted by $\Sigma=(G,\sigma)$, is a graph consisting of an ordinary graph $G$ and a sign function $\sigma : E(G) \rightarrow \{+1,-1\}$ which labels each edge of $G$ as positive or negative. In $\Sigma=(G,\sigma)$, $G$ is called the underlying graph of $\Sigma$ and the set $\sigma^{-1}(-1)=\{e\in E(G)~|~\sigma(e)=-1\}$ is called the signature of $\Sigma.$ Switching $\Sigma$ by a vertex $u$ changes the sign of each edge incident to $u$. Two signed graph $\Sigma_1 =(G,\sigma_1)$ and $\Sigma_2 = (G,\sigma_2)$ are called switching equivalent if one can be obtained by a sequence of switchings from the other.

The following characterization of two signed graphs to be switching equivalent is given by Zaslavsky [5].

[5]
Two signed graphs $\Sigma_1,\Sigma_2$ are switching equivalent if and only if they have the same set of negative cycles.

Given a graph $G$ on $n$ vertices and $m$ edges, there are $2^m$ ways of constructing signed graphs on $G$. An elementary proof of the following lemma is given in <cit.>.

There are $2^{m-n+1}$ switching non-equivalent signed graphs on a connected graph $G$ on $n$ vertices and $m$ edges.

We say the signed graphs $\Sigma_1 =(G,\sigma_1)$ and $\Sigma_2 = (H,\sigma_2)$ are isomorphic if there exists a graph isomorphism between $G$ and $H$ preserving the edge signs. Two signed graphs are switching isomorphic if one is isomorphic to a switching of the other. Two signed graphs $(G,\sigma_1)$ and $(H,\sigma_2)$ are automorphic if they are isomorphic to each other and $G=H$.

Up to switching isomorphism, it is known that there are two signed $K_3$, three signed $K_4$, and seven signed $K_5$. In [3], the authors classified all sixteen switching non-isomorphic signed $K_6$. Mallows and Sloane [2] proved that the number of switching non-isomorphic signed complete graphs on $n$ vertices is equal to the number of Euler graphs on $n$ vertices. In [5], Zaslavsky proved that there are only six signed Petersen graphs, up to switching isomorphism.

Recently, Y. Bagheri et al. [1] proved that the number of mutually switching non-isomorphic signed graphs associated with a given graph $G$ is equal to the number of orbits of the automorphism group of $G$ acting on the set of all possible signed graphs with underlying graph $G$. In this paper, we have used a different technique to determine the number of switching non-isomorphic signed wheels of some particular orders.

A wheel, denoted by $W_n$, is the join of the cycle $C_n$ and a vertex. Let $V(W_n)=\{v,v_1,v_2,...,v_n\}$ and $E(W_n)=\{vv_i, v_iv_{i+1}~|~i=1,2,...,n\}$, where the subscripts are read modulo $n$. For $1 \leq i \leq n$, the edges $vv_i$ are said to be the spokes of $W_n$, and the cycle induced by edges $v_iv_{i+1}$ is said to be the outer cycle, denoted by $C_n$, of $W_n$.

For $n=3$, the graph $W_3$ is the complete graph $K_4$. It is known that the number of switching non-isomorphic signed graphs over $K_4$ is 3. Thus, in the subsequent discussion, we consider the wheels $W_n$ for $n \geq 4$.

For a fixed $0 \leq p \leq n$, $\psi_{p}(n)$ denotes the number of switching non-isomorphic signed wheels with exactly $p$ negative edges on $C_n$. By $\psi(n)$, we denote the number of switching non-isomorphic signed wheels. Thus, $\psi(n) = \sum_{p=0}^{n} \psi_{p}(n)$.

The values of $\psi_{p}(n)$ for $p=0,1,2,3,4,n-4,n-3,n-2,n-1,n$ are determined in Section <ref>. Using these values, the values of $\psi(n)$ for $n \leq 10$ are obtained in Section <ref>.

§ TERMINOLOGY AND METHODOLOGY

Our approach to enumerate the switching non-isomorphic signed wheels is to put $p$ negative edges on $C_n$ at different distances that generate all mutually switching non-isomorphic signed wheels.

By $G_n$, we denote a $regular~n\text{-}gon$ having vertex set $V(G_n)=\{v_{1},v_{2},v_{3},...,v_{n}\}$ and edge set $E(G_n) = \{v_{i}v_{i+1}~|~i=1,2,...,n\}$, where the subscripts are read modulo $n$.

The distance between two vertices $u~\text{and}~v$, denoted $d(u,v)$, in a graph $G$ is defined to be the number of edges in a shortest path between $u~\text{and}~v$. In $G_n$, it is clear that $1 \leq d(v_i,v_j) \leq \lfloor \frac{n}{2} \rfloor$ for all $i \neq j$. Further, if we measure the distance along one particular direction (in clockwise or anticlockwise), then we have $1 \leq d(v_i,v_j) \leq n-1$ for all $i \neq j$.

If $n$ is an even number then the vertices $v_{i}$ and $v_{i+ \frac{n}{2}}$ are called diagonally opposite vertices and the edges $v_{i}v_{i+1}$ and $ v_{i+ \frac{n}{2}}v_{i+1+\frac{n}{2}}$ are called the opposite edges. On the other hand, if $n$ is an odd number, for any $v_{i}$, the edge $v_{i+\lfloor \frac{n}{2} \rfloor}v_{i+(\lfloor \frac{n}{2} \rfloor+1)}$ is called the opposite edge of $v_{i}$, $1 \leq i \leq n$.

Clearly, $G_n$ features $n$ axes of symmetry. A common point at which all these axes meet is called the center of $G_n$. Observe that if $n$ is an even number then half of the axes pass through diagonally opposite vertices and the remaining axes pass through the midpoints of opposite edges. On the other hand, if $n$ is an odd number, all the axes pass through a vertex and the midpoint of its opposite edge.

Let Aut($G$) denotes the automorphism group of a graph $G$. It is well known that Aut($W_n$) = Aut($G_n$)  = $<\alpha, \beta~|~ \alpha^n = \beta^2 = 1, \alpha\beta = \beta\alpha^{-1}>$, the dihedral group $D_n$.

If a spoke $vv_{j}$, for some $1\leq j \leq n$, is negative in $(W_n, \sigma)$ then one can make it positive by switching by $v_j$. Thus for any $(W_n, \sigma)$ there is an equivalent $(W_n, \sigma_1)$ such that $\sigma_1^{-1}(-1) \subseteq E(C_{n})$. Therefore, in the subsequent discussion, we only consider the signed wheels whose negative edges lie on $C_n$.

The following result will be helpful to examine whether two signed wheels are switching equivalent.

Two signed wheels with different signatures are always switching non-equivalent.

Let $\Sigma_1=(W_n,\sigma_1)$ and $\Sigma_2=(W_n,\sigma_2)$ be two signed wheels such that $\sigma_{1}^{-1}(-1) \neq \sigma_{2}^{-1}(-1)$. Since each negative edge makes exactly one triangle negative, the result follows directly from Lemma <ref>.

Let $\Sigma=(W_n,\sigma)$ be a signed wheel with $p$ negative edges. Corresponding to $\Sigma$, we associate an ordered distance tuple $D(\Sigma)=(r_{0},r_{1},r_{2},r_{3},...,r_{\lfloor \frac{n}{2} \rfloor})$, where $r_{l}$ denotes the number of distinct pairs of negative edges which are at distance $l$ and $r_{0}+r_{1}+r_{2}+r_{3}+\cdots +r_{\lfloor \frac{n}{2} \rfloor} = \binom{p}{2}$.

Consider $\Sigma = (W_8,\sigma)$, as depicted in Figure <ref>. Let $e_1= v_1v_2, e_2=v_2v_3, e_3=v_4v_5, e_4=v_7v_8$ so that $\sigma^{-1}(-1)= \{e_1,e_2,e_3,e_4\}$. It is easy to see that $d(e_1,e_2)=0,~d(e_1,e_3)=2,~d(e_1,e_4)=1,~d(e_2,e_3)=1,~\text{dist}(e_2,e_4)=2,~d(e_3,e_4)=2$. Therefore, $r_{0}=1,~r_{1}=2,~r_{2}=3,~r_{3}=0,~r_{4}=0$. Hence we have $D(\Sigma) = (1,2,3,0,0)$.

[vertex] (v1) at (10,0) ;
[vertex] (v2) at (11.50,0.90) ;
[vertex] (v3) at (12.5,2.5) ;
[vertex] (v4) at (11.50,4.15) ;
[vertex] (v5) at (10,5) ;
[vertex] (v6) at (8.25,4.15) ;
[vertex] (v7) at (7.5,2.5) ;
[vertex] (v8) at (8.50,0.90) ;
[vertex] (v9) at (10,2.5) ;

[above] at (v5) $v_{1}$;
[xshift=0.25cm, yshift=0.15cm] at (v4) $v_{2}$;
[xshift=0.3cm, yshift=-0.05cm] at (v3) $v_{3}$;
[xshift=0.25cm, yshift=-0.2cm] at (v2) $v_{4}$;
[xshift=0.05cm, yshift=-0.3cm] at (v1) $v_{5}$;
[xshift=-0.25cm, yshift=-0.2cm] at (v8) $v_{6}$;
[left] at (v7) $v_{7}$;
[xshift=-0.25cm, yshift=0.15cm] at (v6) $v_{8}$;
at (10.2,3) $v$;

/→in v2/v3,v5/v6,v7/v8,v8/v1,v9/v1,v9/v2,v9/v3,v9/v4,v9/v5,v9/v6,v9/v7,v9/v8 () – (→);

[dashed] (10,0) – (11.50,0.90);
[dashed] (10,5) – (11.50,4.15);
[dashed] (12.5,2.5) – (11.50,4.15);
[dashed] (7.5,2.5) – (8.25,4.15);

A signed $W_8$

The following lemma will help us in deciding whether two signed wheels with $p$ negative edges are automorphic.

Two signed wheels $\Sigma_1$ and $\Sigma_2$ with $p$ negative edges are automorphic to each other if and only if $D(\Sigma_1) = D(\Sigma_2)$.

Let $\Sigma_1$ and $\Sigma_2$ be automorphic to each other. Since an automorphism of $W_n$ preserve the distance, it follows that $D(\Sigma_1) = D(\Sigma_2)$.

Conversely, let $\Sigma_1$ and $\Sigma_2$ be two signed wheels with $p$ negative edges such that $D(\Sigma_1) = D(\Sigma_2)$. We need to show that $\Sigma_1$ and $\Sigma_2$ are automorphic to each other. To establish an automorphism of $W_n$ that maps $\Sigma_1$ onto $\Sigma_2$, we first fix the position of $p$ negative edges of $\Sigma_1$ in clockwise direction, say, at $v_{1_1}v_{1_1 +1}, v_{1_2}v_{1_2 +1},v_{1_3}v_{1_3 +1},...,v_{1_p}v_{1_p +1}$ such that $1 \leq 1_i < 1_j \leq n$ for $1 \leq i < j \leq p$.

Since $D(\Sigma_1) = D(\Sigma_2)$, the positions of $p$ negative edges of $\Sigma_2$ can also be fixed in clockwise direction say, at $v_{2_1}v_{2_1+1}, v_{2_2}v_{2_2+1},v_{2_3}v_{2_3 +1},...,v_{2_p}v_{2_p +1}$, where $1 \leq 2_{i} \leq n$ and subscripts are read modulo $n$, so that

\begin{equation}\label{equation 1}
d(v_{1_i}v_{1_i +1},v_{1_j}v_{1_j +1})=d(v_{2_i}v_{2_i+1},v_{2_j}v_{2_j +1}),~\text{ for all}~ i,j\in \{1,2,...,p\}.
\end{equation}

Define $\phi : V(W_n) \rightarrow V(W_n)$ by

$$\phi(x) = \begin{cases} v & \text{if $x=v$} \\
v_{2_1 + t} & \text{if $x=v_{1_1 + t}$ for $t=0,1,2,...,n-1$}.
\end{cases}$$

It is easy to verify that $\phi$ is an automorphism of $W_n$ that maps $\Sigma_1$ onto $\Sigma_2$. Hence if $D(\Sigma_1) = D(\Sigma_2)$ then $\Sigma_1$ and $\Sigma_2$ are automorphic to each other.

Lemma <ref> and Lemma <ref> together yields the following result.

Let $\Sigma_1$ and $\Sigma_2$ be two signed wheels with $p$ negative edges such that $D(\Sigma_1) \neq D(\Sigma_2)$. Then $\Sigma_1$ and $\Sigma_2$ are switching non-isomorphic.

Among any four edges $e_1, e_2, e_3~\text{and}~e_4$ of $C_n$, there exist two edges $e_i$ and $e_j$ such that $\text{d}(e_i,e_j) \leq \lfloor \frac{n-4}{4} \rfloor$.

For a fix $n$, let if possible

\begin{equation}
d(e_i,e_j) \geq \lfloor \frac{n-4}{4} \rfloor + 1,~\text{for all}~ i,j \in \{1,2,3,4\},~ i \neq j.
\end{equation}

Note that if the distance between $e_i~\text{and}~e_j$ is $k$ then there are at least $k-1$ vertices between end vertices of $e_i~\text{and}~e_j$. Therefore there are at least $\lfloor \frac{n-4}{4} \rfloor$ vertices between $e_i~\text{and}~e_j~\text{for all}~ i,j \in \{1,2,3,4\}$. This means there are at least $4\lfloor \frac{n-4}{4} \rfloor +8$ vertices in $C_n$, a contradiction. Hence the result follows.

Let $e_1,e_2,e_3~\text{and}~e_4$ be four negative edges which lie on $C_n$. We place the edges $e_1,e_2,e_3~\text{and}~e_4$ in such a way that if $i < j$ and $e_i=v_{r}v_{r+1}, e_j=v_{l}v_{l+1}$ then $r+1 \leq l$. Further, in light of Lemma <ref>, we can always assume that $\text{d}(e_1,e_2) \leq \lfloor \frac{n-4}{4} \rfloor$. Without loss of generality, let $e_1=v_1v_2$. To calculate the value of $\psi_{4}(n)$, we will count the different signatures of size four by applying the following strategies.

S1. Take $d(e_1,e_2)=0$ and count the different possibilities for $e_3~\text{and}~e_4$ up to automorphisms. This is carried out in Lemma <ref>, Lemma <ref>, Lemma <ref> and Lemma <ref>.

S2. Take $d(e_1,e_2)=1$ and count the choices for $e_3~\text{and}~e_4$ under the following conditions:

(i) $d(e_2,e_3) \geq 1$;

(ii) $d(e_3,e_4) \geq 1$;

(iii) $d(e_4,e_1) \geq 1$.

Note that if any one of $d(e_2,e_3),d(e_3,e_4),d(e_4,e_1)$ is zero then replacement of those two edges with $e_1~\text{and}~e_2$ will give us a signature which is already encountered in S1.

S3. For $d(e_1,e_2)= r$, where $1 \leq r < \lfloor \frac{n-4}{4} \rfloor$, count different choices of $e_3~\text{and}~e_4$.

S4. If $d(e_1,e_2)= r+1$, where $2 \leq r+1 \leq \lfloor \frac{n-4}{4} \rfloor$, count the different choices for $e_3~\text{and}~e_4$ under the following conditions:

(i) $d(e_2,e_3) \geq r+1$;

(ii) $d(e_3,e_4) \geq r+1$;

(iii) $d(e_4,e_1) \geq r+1$.

Note that if any one of $d(e_2,e_3),d(e_3,e_4),d(e_4,e_1)$ is less than $r+1$ then replacement of those two edges with $e_1~\text{and}~e_2$ will give us a signature which is already encountered in S3.

§ COMPUTATION

In this section, we compute the value of $\psi_{p}(n)$ for $p=0,1,2,3,4,n-4,n-3,n-2,n-1,~\text{and}~n$, where $n \geq 4$. To count the number of switching non-isomorphic signed wheels with $p$ negative edges, it is enough to count the different choices of $p$ edges from $E(C_n)$ up to automorphisms. Note that the counting of different $p$ edges on $C_n$ is same as the counting of different $n-p$ edges. Thus for any $0 \leq p \leq n$, we have

\begin{equation}\label{main equation}
\psi_{p}(n) = \psi_{n-p}(n).
\end{equation}

The following lemma is trivial.

For each $n \geq 4$, $\psi_{0}(n)= \psi_{n}(n) =1$.

Any two signed wheels with exactly one negative edge are automorphic to each other. Therefore, in the view of Equation <ref>, the following lemma is immediate.

For each $n \geq 4$, $\psi_{1}(n)=\psi_{n-1}(n)=1$.

We now determine the value of $\psi_{n-2}(n)~\text{and}~\psi_{2}(n)$.

For each $n \geq 4$, $\psi_{2}(n)=\psi_{n-2}(n)=1+\lfloor \frac{n-2}{2} \rfloor$.

We classify it into two cases:

Case 1. If two edges form a path $P_{3}$ then there is only one possibility up to rotations. One such path is $P_{3}= v_{1}v_{2}v_{3}$.

Case 2. If two edges are disjoint, then the number of choices is $\lfloor \frac{n-2}{2} \rfloor$ up to automorphisms.

Each choice of two edges in Case 1 and Case 2 produces a signed wheel with two negative edges. In light of Lemma <ref>, all these signed wheels are mutually switching non-isomorphic. This proves that $\psi_{2}(n)=\psi_{n-2}(n)=1+\lfloor \frac{n-2}{2} \rfloor$.

A number $n$ is said to have a $k\text{-}partition$ if $n=\lambda_1+\lambda_{2}+...+\lambda_{k}$, where we assume $\lambda_{1} \geq \lambda_{2} \geq \lambda_{3} \geq ...\geq\lambda_{k} \geq 1$. Par($n;k$) denotes the set of all $k$-partitions of $n$ with $p(n;k) = |\text{Par}(n;k)|$. Clearly, the number $p(n;k)$ is zero if $n<k$. The number $p(n-3;3)$ is used to compute $\psi_{n-3}(n)$.

For each $n \geq 4$, $\psi_{3}(n)=\psi_{n-3}(n)=1+\lfloor \frac{n-3}{2} \rfloor +p(n-3;3)$.

Since $n-3$ edges are to be chosen from $C_n$, only following three cases are possible:

(i) all $n-3$ edges form a path;

(ii) $n-3$ edges form two different paths;

(iii) $n-3$ edges form three different paths.

Clearly, there is only one possibility in case (i), up to rotations. For case (ii), the number of two different paths comprising $n-3$ edges is same as the number of partitions of $n-3$ with exactly two parts. Therefore, the number of two different paths is $\lfloor \frac{n-3}{2} \rfloor$.

For case (iii), let three distinct paths formed by $n-3$ edges be $P_{t}, P_{t^\prime},~\text{and}~P_{t^{\prime\prime}}$ such that $t \geq t^{\prime}\geq t^{\prime\prime} \geq 2$. For each $t \geq t^{\prime}\geq t^{\prime\prime} \geq 2$, it is easy to see that there is a unique possibility for three such paths, up to rotations. Thus the number of three such paths is same as the number of partitions of $n-3$ with exactly 3 parts. Hence there are $p(n-3;3)$ different choices for three such paths.

Each different possibility of $n-3$ edges in case (i), (ii), and (iii) produces a signed wheel with $n-3$ negative edges and in light of Lemma <ref>, all these signed wheels are mutually switching non-isomorphic. Hence $\psi_{n-3}(n)=1+\lfloor \frac{n-3}{2} \rfloor +p(n-3;3)=\psi_{3}(n)$.

Let $\Sigma=(W_n, \sigma)$ be a signed wheel with exactly four negative edges $e_1, e_2, e_3,~\text{and}~e_4$ on $C_n$. By Lemma <ref>, it is possible to choose two edges $e_i$ and $e_j$ so that $\text{d}(e_i,e_j) \leq \lfloor \frac{n-4}{4} \rfloor$. Again, a rotation permits us to choose these two edges as $e_1$ and $e_2$ so that $\text{d}(e_1,e_2) \leq \lfloor \frac{n-4}{4} \rfloor$. We now proceed to compute $\psi_{4}(n)$ and to do so we will make use of S1, S2, S3 and S4 .

If edges $e_1, e_2, e_3~\text{and}~e_4$ form a path on $C_n$, then there is only one signed wheel up to rotation.

If edges $e_1, e_2~\text{and}~e_3$ form a path $P_4$ and the edge $e_4$ is at distance at least one from $P_4$, then the number of non-automorphic signed wheels is $ \lfloor \frac{n}{2} \rfloor -2$.

Let $e_1=v_1v_2, e_2=v_2v_3~\text{and}~e_3=v_3v_4$. Due to the reflection passing through the mid point of the edge $e_2$, the edge $e_4$ can be $v_{5}v_{6}, v_{6}v_{7},..., v_{\lfloor \frac{n}{2} \rfloor+2}v_{\lfloor \frac{n}{2} \rfloor +3}$ for a total of $\lfloor \frac{n}{2} \rfloor -2$.

If edges $e_1, e_2$ form a path $P_3$ and $e_3,e_4$ form an another path on three vertices disjoint from $P_3$, then the number of non-automorphic signed wheels is $ \lfloor \frac{n}{2} \rfloor -2$.

Let $e_1=v_1v_2, e_2=v_2v_3$ and $P_3= v_1v_2v_3$. Let $e_3~\text{and}~e_4$ form an another path $P_3^{\prime}$ different from $P_3$. Due to the reflection passing through $v_2$, the path $P_3^{\prime}$ can be $v_{4}v_{5}v_{6}, v_{5}v_{6}v_{7},...,v_{\lfloor \frac{n}{2} \rfloor +1}v_{\lfloor \frac{n}{2} \rfloor +2}v_{\lfloor \frac{n}{2} \rfloor +3}$ for a total of $\lfloor \frac{n}{2} \rfloor -2$.

Let edges $e_1, e_2$ form a path $P_3$ and $e_3,e_4$ be non-adjacent with each other as well as with $P_3$. Then the number of non-automorphic signed wheels is $(k-2)^2$ and $(k-3)(k-2)$ when $n=2k+1$ and $n=2k$, respectively.

Let $e_1=v_1v_2, e_2=v_2v_3$ and $P_3= v_1v_2v_3$. We classify $n$ into two cases:

Case 1. Let $n=2k+1$. If $e_3=v_4v_5$ then due to the reflection passing through $v_{2}$, the edge $e_4$ can be $v_{6}v_{7}, v_{7}v_{8},...,v_{2k}v_{2k+1}$ for a total of $2k-5$.

If $e_3=v_{l}v_{l+1}$, for $5 \leq l \leq k+1$, then the edge $e_4$ can be $v_{l+2}v_{l+3},...,v_{2k-l+4}v_{2k-l+5}$ for a total of $2k-2l+3$. The number of different choices of $e_3$ and $e_4$ is

\begin{align*}
& (2k-5)+ \sum_{l=5}^{k+1}[2k-2l+3] \\
=& (2k-5)+2k(k-3)-2[\frac{(k+1)(k+2)}{2}- 10]+3(k-3)\\
=& k^2-4k+4\\
=& (k-2)^2.
\end{align*}

Case 2. Let $n=2k$. If $e_3=v_{l}v_{l+1}$, for $4 \leq l \leq k$, then the edge $e_4$ can be $v_{l+2}v_{l+3},...,v_{2k-l+3}v_{2k-l+4}$ for a total of $2k-2l+2$. The number of different choices of $e_3$ and $e_4$ is

\begin{align*}
& \sum_{l=4}^{k}[2k-2l+2] \\
=& 2k(k-3)-2[\frac{(k)(k+1)}{2}- 6]+2(k-3)\\
=& k^2-5k+6\\
=& (k-3)(k-2).
\end{align*}

In Case 1 and Case 2, each choice of $e_3$ and $e_4$ along with $P_3$ produces a signed wheel with four negative edges. By Lemma <ref>, all these signed wheels are pairwise non-automorphic. This completes the proof.

Let $(W_{2k+1}, \sigma)$ be a signed wheel with four negative edges in which $d(e_1,e_2)=r$, where $1 \leq r \leq \lfloor \frac{n-4}{4} \rfloor$. Then the number of non-automorphic signed wheels is $[k-(2r+1)]^2$.

Let $e_1=v_1v_2$ and $e_2=v_{r+2}v_{r+3}$ such that $d(e_1,e_2)=r$, where $1 \leq r \leq \lfloor \frac{n-4}{4} \rfloor$. We count the choices for $e_3$ and $e_4$ in the following two cases:

(i) If $e_3=v_{2r+3}v_{2r+4}$, then due to the reflection passing through the mid-point of $e_2$, the edge $e_4$ can be $v_{3r+4}v_{3r+5},...,v_{k+r+2}v_{k+r+3}$ for a total of $k-(2r+1).$

(ii) If $e_3=v_{l}v_{l+1}$, then $e_4$ can be $v_{l+1+r}v_{l+1+r+1},...,v_{(2k+1)-(l-r-3)}v_{(2k+1)-(l-r-3)+1}$ for a total of $2k-2l+4$, where $2r+4 \leq l \leq k+1$.

Thus if $e_1=v_1v_2$, $e_2=v_{r+2}v_{r+3}$, then the number of choices for $e_3$ and $e_4$ is the sum of all choices obtained in (i) and (ii). Each such choice produces a signed wheel with four negative edges and by Lemma <ref>, all these signed wheels are mutually non-automorphic. Hence the number of non-automorphic signed wheels is

\begin{align*}
& k-(2r+1)+ \sum_{l=2r+4}^{k+1} (2k-2l+4) \\
=& \{k-(2r+1)\}+ \{2k(k+1-2r-3)-2\big[\frac{(k+1)(k+2)}{2}-\frac{(2r+3)(2r+4)}{2}\big]+4(k+1-2r-3)\}\\
\end{align*}

This completes the proof.

Let $(W_{2k}, \sigma)$ be a signed wheel with four negative edges in which $d(e_1,e_2)=r$, where $1 \leq r \leq \lfloor \frac{n-4}{4} \rfloor$. Then the number of non-automorphic signed wheels is $[k-(2r+1)]+[k-(2r+2)]^2$.

Let $e_1=v_1v_2$ and $e_2=v_{r+2}v_{r+3}$ such that $d(e_1,e_2)=r$, where $1 \leq r \leq \lfloor \frac{n-4}{4} \rfloor$. We count the different choices for $e_3$ and $e_4$ in the following two cases:

(i) If $e_3=v_{2r+3}v_{2r+4}$, then due to the reflection passing through the mid-point of $e_2$, the edge $e_4$ can be $v_{3r+4}v_{3r+5},...,v_{k+r+2}v_{k+r+3}$ for a total of $k-(2r+1).$

(ii) If $e_3=v_{l}v_{l+1}$, then $e_4$ can be $v_{l+1+r}v_{l+1+r+1},...,v_{(2k)-(l-r-3)}v_{(2k)-(l-r-3)+1}$ for a total of $2k-2l+3$, where $2r+4 \leq l \leq k+1$.

Thus the number of non-automorphic signed wheels is

\begin{align*}
& k-(2r+1)+ \sum_{l=2r+4}^{k+1} (2k-2l+3) \\
=& \{k-(2r+1)\}+ \{2k(k-2r-2)-2[\frac{(k+1)(k+2)}{2}-\frac{(2r+3)(2r+4)}{2}]+3(k-2r-2)\}\\
=& [k-(2r+1)]+[k-(2r+2)]^2.
\end{align*}

This proves the lemma.

Note that, in light of Lemma <ref>, all the signed wheels counted in Lemma <ref> to Lemma <ref> are switching non-isomorphic. We now compute $\psi_{4}(n)$ by classifying $n$ into two cases depending upon whether $n$ is odd or even. By $\psi_{4}^{e}(n)$ and $\psi_{4}^{o}(n)$, we denote the number of switching non-isomorphic signed wheels with exactly four negative edges, when $n$ is even and odd, respectively. In the following two theorems, we put $ \lfloor \frac{n-4}{4} \rfloor=l$.

Let $n = 2k$, for some $k \geq 2$. Then

\begin{equation}\label{n even and n/4 even}
\psi_{4}^{e}(n) =  (l+1)k^{2} - (2l+3)(l+1)k +  \frac{4l^3+15l^2+20l+9}{3}.
\end{equation}

Let $\psi_{i}$ be the number of non-automorphic signed wheels with four negative edges $e_1,e_2,e_3,e_4$ such that $d(e_1,e_2) = i$, where $0 \leq i \leq l$. Thus we have

\begin{align*}
\psi_{4}^{e}(n) &= \sum_{i=0}^{l} \psi_{i} \\
&= \{1+(k-2)+(k-2)+(k-3)(k-2)\} + \sum_{i=1}^{l} \psi_{i}\\
&= \{k^2-3k+3\} + \sum_{i=1}^{l} [k-(2i+1)]+[k-(2i+2)]^2 \\
&= \{k^2-3k+3\} + \sum_{i=1}^{l} [k^2-3k+6i-4ki+4i^2+3]\\
&= \{k^2-3k+3\} + \Big \{ lk^2-3kl+6\frac{l(l+1)}{2}-4k \frac{l(l+1)}{2}+4 \frac{l(l+1)(2l+1)}{6} +3l \Big \} \\
&= (l+1)k^2-(2l+3)(l+1)k +  \frac{4l^3+15l^2+20l+9}{3}.
\end{align*}

This completes the proof.

Note that the value of $\psi_{0}$ is the sum of all the values obtained in Lemma <ref>, <ref>, <ref> and Lemma <ref>. For each $1 \leq i \leq l$, the value of $\psi_{i}$ is given in Lemma <ref>.

Let $n = 2k+1$, for some $k \geq 2$. Then

\begin{equation}\label{n odd and n/4 even}
\psi_{4}^{o}(n) =  (l+1)k^{2} - 2(l+1)^2k +  \frac{(2l+1)(2l+3)(l+1)}{3}.
\end{equation}

Let $\psi_{i}$ be the number defined in the proof of the Theorem <ref>.  Thus we have

\begin{align*}
\psi_{4}^{o}(n) &= \sum_{i=0}^{l} \psi_{i} \\
&= \{1+(k-2)+(k-2)+(k-2)^2\} + \sum_{i=1}^{l} \psi_{i}\\
&= \{k^2-2k+1\} + \sum_{i=1}^{l} [k-(2i+1)]^2 \\
&= \{k^2-2k+1\} + \sum_{i=1}^{l} [k^2-2k+4i-4ki+4i^2+1]\\
&= \{k^2-2k+1\} + \Big \{ lk^2-2kl+4\frac{l(l+1)}{2}-4k \frac{l(l+1)}{2}+4 \frac{l(l+1)(2l+1)}{6} +l \Big \} \\
&= (l+1)k^{2} - 2(l+1)^2k +  \frac{(2l+1)(2l+3)(l+1)}{3}.
\end{align*}

This completes the proof.

§ MAIN RESULTS

In this section, we compute the number of switching non-isomorphic signed wheels $W_{n}$, for $4 \leq n \leq 10$.

The value of $\psi_{5}(10)$ is 15.

To count $\psi_{5}(10)$, the different choices for five edges on $C_{10}$ are considered in the following cases.

* If the five edges form a path $P_6$, then there is only one choice for such a path, up to rotation.

* If the set of five edges is a disjoint union of $P_5~\text{and}~P_2$ then we can assume that $P_5=v_1v_2v_3v_4v_5$. Due to the reflection passing through $v_3$ and $v_8$, the possibilities for $P_2$ are $v_6v_7$ and $v_7v_8$. Therefore there are only two such choices.

* If the set of five edges is a disjoint union of $P_4~\text{and}~P_3$, we assume that $P_4=v_1v_2v_3v_4$. Due to the reflection passing through the mid points of $v_2v_3$ and its opposite edge $v_7v_8$, the choices for $P_3$ are $v_5v_6v_7$ or $v_6v_7v_8$. Thus there are only two such choices.

* If the set of five edges is a disjoint union of $P_4,P_2^{1}~\text{and}~P_2^{2}$, where $P_2^{1}~\text{and}~P_2^{2}$ are paths on two vertices, we assume that $P_4=v_1v_2v_3v_4$. Further, if $P_2^{1}=v_5v_6$, then $P_2^{2}$ can be $v_{7}v_{8},v_{8}v_{9},v_{9}v_{10}$. If $P_2^{1}=v_6v_7$, then $P_2^{2}$ must be $v_8v_9$. Hence there are four such choices.

* If the set of five edges of is a disjoint union of $P_3^{1},P_3^{2}~\text{and}~P_2$, where $P_3^{1},P_3^{2}$ are paths on three vertices, we assume that $P_3^{1}=v_1v_2v_3$. If $P_3^{2}=v_4v_5v_6$ then due to the reflection passing through the mid points of $v_3v_4$ and its opposite edge $v_8v_9$, $P_2$ can be $v_7v_8~\text{or}~ v_8v_9$. If $P_3^{2}=v_5v_6v_7$ then due to the reflection passing through $v_4$ and its opposite vertex $v_9$, $P_2$ must be $v_8v_9$. Finally, if $P_3^{2}=v_6v_7v_8$ then due to the reflection passing through mid points of $v_4v_5$ and its opposite edge $v_9v_{10}$, $P_2$ must be either $v_4v_5$ or $v_9v_{10}$. Thus there are four choices for this case.

* If the set of five edges is a disjoint union of $P_3,P_2^{1},P_2^{2}~\text{and}~P_2^{3}$, where $P_2^{1},P_2^{2}~\text{and}~P_2^{3}$ are paths on two vertices, then there are two such choices, up to automorphisms.

From all the cases considered, we find that $\psi_{5}(10)=15$. These 15 signed $W_{10}$ are shown in Figure <ref>.

For $4 \leq n \leq 10$ and $0 \leq p \leq 10$, the values of $\psi_{p}(n)$ are those listed in Table <ref>.

In Table <ref>, entries of row $i$, for $i=2,3,4,~\text{and}~5$, are computed from Lemma <ref>, <ref>, <ref>, and Lemma <ref> respectively. The values of $\psi_{r}(s)$ for $r=s$ are computed from Lemma <ref> and of $\psi_{r}(s)$ for $r=s-1$ are computed from Lemma <ref>. The values of $\psi_{r-2}(r)$ and of $\psi_{r-3}(r)$ for $r=7,8,9,~\text{and}~10$ are computed from Lemma <ref> and Lemma <ref>, respectively. The values of $\psi_{4}(8)$ and $\psi_{4}(10)(=\psi_{6}(10))$ are computed from Theorem <ref> and of $\psi_{4}(9)(=\psi_{5}(9))$ is computed from Theorem <ref>. The value of $\psi_{5}(10)$ is obtained in Lemma <ref>. This proves the lemma.

For $n=4,5,6,7,8,9,10$, the number of switching non-isomorphic signed wheels on $W_n$ are those given in Table <ref>.

The values of Table <ref> are obtained by respective columns sums of Table <ref>.

$p$$n$     4     5     6     7     8     9     10

0     1     1     1     1     1     1     1

1      1     1     1     1     1     1     1

2     2     2     3     3     4     4     5

3     1     2     3     4     5     7     8

4     1     1     3     4     8     10     16

5         1     1     3     5     10     15

6               1     1     4     7     16

7                    1     1     4     8

8                         1     1     5

9                              1     1

10                                   1

The number $\psi_{p}(n)$, for $n=4,5,...,10$ and $0 \leq p \leq 10$

$n$      4     5     6     7     8     9     10

$\psi(n)$     6     8     13     18     30     46     77

The number of switching non-isomorphic signed $W_n$, for $n=4,5,...,10$

§ CONCLUSION

Recall from Lemma <ref> that the number of switching non-equivalent signed wheels are $2^{n}$. Another way of getting this number is the following.

It was already noticed that any signed wheel is switching equivalent to a signed wheel whose signature is a subset of $E(C_n)$. Also, by Lemma <ref>, any two signed wheels whose signatures are different subsets of $E(C_n)$ are switching non-equivalent. As the total number of subsets of $E(C_n)$ are $2^{n}$, there are $2^{n}$ switching non-equivalent signed wheels on $n+1$ vertices. However many of these $2^{n}$ signed wheels are isomorphic to each other. For this purpose, we have determined the value of $\psi_{p}(n)$, for $p=0,1,2,3,4,n-4,n-3,n-2,n-1,n$ and the value of $\psi(n)$, for $n=4,5,6,7,8,9,10$. The values of $\psi_{p}(n)$, for $p=5,6,...,n-5$ are still unknown.

[vertex] (v0) at (360/10*0 :0.1cm) ;
[vertex] (v3) at (360/10*0 :7cm) ;
[vertex] (v2) at (360/10*1 :7cm) ;
[vertex] (v1) at (360/10 *2 :7cm) ;
[vertex] (v10) at (360/10 *3 :7cm) ;
[vertex] (v9) at (360/10 *4 :7cm) ;
[vertex] (v8) at (360/10 *5:7cm) ;
[vertex] (v7) at (360/10 *6 :7cm) ;
[vertex] (v6) at (360/10 *7:7cm) ;
[vertex] (v5) at (360/10 *8:7cm) ;
[vertex] (v4) at (360/10 *9:7cm) ;
[dashed] (360/10 *0 +1.5:7cm) arc  (360/10*0+1.5:360/10*1 -1.5 :7cm);
[dashed] (360/10 *1 +1.5:7cm) arc  (360/10*1+1.5:360/10*2 -1.5 :7cm);
(360/10 *2 +1.5:7cm) arc  (360/10*2+1.5:360/10*3 -1.5 :7cm);
(360/10 *3 +1.5:7cm) arc  (360/10*3+1.5:360/10*4 -1.5 :7cm);
(360/10 *4 +1.5:7cm) arc  (360/10*4+1.5:360/10*5 -1.5 :7cm);
(360/10 *5 +1.5:7cm) arc  (360/10*5+1.5:360/10*6 -1.5 :7cm);
(360/10 *6 +1.5:7cm) arc  (360/10*6+1.5:360/10*7 -1.5 :7cm);
[dashed] (360/10 *7 +1.5:7cm) arc  (360/10*7+1.5:360/10*8 -1.5 :7cm);
[dashed] (360/10 *8 +1.5:7cm) arc  (360/10*8+1.5:360/10*9 -1.5 :7cm);
[dashed] (360/10 *9 +1.5:7cm) arc  (360/10*9+1.5:360/10*10 -1.5 :7cm);

[xshift=0.2cm, yshift=0.15cm] at (360/10*0 :1.5cm) $v$;
[xshift=0.25cm] at (360/10*0 :7cm) $v_3$;
[xshift=0.25cm, yshift=0.1cm] at (360/10*1 :7cm) $v_2$;
[xshift=0.1cm, yshift=0.25cm] at (360/10 *2 :7cm) $v_1$;
[xshift=-0.1cm, yshift=0.25cm] at (360/10 *3 :7cm) $v_{10}$;
[xshift=-0.2cm, yshift=0.15cm] at (360/10 *4 :7cm) $v_9$;
[xshift=-0.25cm] at (360/10 *5:7cm) $v_8$;
[xshift=-0.25cm, yshift=-0.15cm] at (360/10 *6 :7cm) $v_7$;
[xshift=-0.1cm, yshift=-0.3cm] at (360/10 *7:7cm) $v_6$;
[xshift=0.05cm, yshift=-0.3cm] at (360/10 *8:7cm) $v_5$;
[xshift=0.2cm, yshift=-0.2cm] at (360/10 *9:7cm) $v_4$;
/→in v0/v1,v0/v2,v0/v3,v0/v4,v0/v5,v0/v6,v0/v7,v0/v8,v0/v9,v0/v10 () – (→);

[vertex] (v0) at (360/10*0 :0.1cm) ;
[vertex] (v3) at (360/10*0 :7cm) ;
[vertex] (v2) at (360/10*1 :7cm) ;
[vertex] (v1) at (360/10 *2 :7cm) ;
[vertex] (v10) at (360/10 *3 :7cm) ;
[vertex] (v9) at (360/10 *4 :7cm) ;
[vertex] (v8) at (360/10 *5:7cm) ;
[vertex] (v7) at (360/10 *6 :7cm) ;
[vertex] (v6) at (360/10 *7:7cm) ;
[vertex] (v5) at (360/10 *8:7cm) ;
[vertex] (v4) at (360/10 *9:7cm) ;
[dashed] (360/10 *0 +1.5:7cm) arc  (360/10*0+1.5:360/10*1 -1.5 :7cm);
[dashed] (360/10 *1 +1.5:7cm) arc  (360/10*1+1.5:360/10*2 -1.5 :7cm);
(360/10 *2 +1.5:7cm) arc  (360/10*2+1.5:360/10*3 -1.5 :7cm);
(360/10 *3 +1.5:7cm) arc  (360/10*3+1.5:360/10*4 -1.5 :7cm);
(360/10 *4 +1.5:7cm) arc  (360/10*4+1.5:360/10*5 -1.5 :7cm);
(360/10 *5 +1.5:7cm) arc  (360/10*5+1.5:360/10*6 -1.5 :7cm);
[dashed] (360/10 *6 +1.5:7cm) arc  (360/10*6+1.5:360/10*7 -1.5 :7cm);
(360/10 *7 +1.5:7cm) arc  (360/10*7+1.5:360/10*8 -1.5 :7cm);
[dashed] (360/10 *8 +1.5:7cm) arc  (360/10*8+1.5:360/10*9 -1.5 :7cm);
[dashed] (360/10 *9 +1.5:7cm) arc  (360/10*9+1.5:360/10*10 -1.5 :7cm);

[xshift=0.2cm, yshift=0.15cm] at (360/10*0 :1.5cm) $v$;
[xshift=0.25cm] at (360/10*0 :7cm) $v_3$;
[xshift=0.25cm, yshift=0.1cm] at (360/10*1 :7cm) $v_2$;
[xshift=0.1cm, yshift=0.25cm] at (360/10 *2 :7cm) $v_1$;
[xshift=-0.1cm, yshift=0.25cm] at (360/10 *3 :7cm) $v_{10}$;
[xshift=-0.2cm, yshift=0.15cm] at (360/10 *4 :7cm) $v_9$;
[xshift=-0.25cm] at (360/10 *5:7cm) $v_8$;
[xshift=-0.25cm, yshift=-0.15cm] at (360/10 *6 :7cm) $v_7$;
[xshift=-0.1cm, yshift=-0.3cm] at (360/10 *7:7cm) $v_6$;
[xshift=0.05cm, yshift=-0.3cm] at (360/10 *8:7cm) $v_5$;
[xshift=0.2cm, yshift=-0.2cm] at (360/10 *9:7cm) $v_4$;
/→in v0/v1,v0/v2,v0/v3,v0/v4,v0/v5,v0/v6,v0/v7,v0/v8,v0/v9,v0/v10 () – (→);

[vertex] (v0) at (360/10*0 :0.1cm) ;
[vertex] (v3) at (360/10*0 :7cm) ;
[vertex] (v2) at (360/10*1 :7cm) ;
[vertex] (v1) at (360/10 *2 :7cm) ;
[vertex] (v10) at (360/10 *3 :7cm) ;
[vertex] (v9) at (360/10 *4 :7cm) ;
[vertex] (v8) at (360/10 *5:7cm) ;
[vertex] (v7) at (360/10 *6 :7cm) ;
[vertex] (v6) at (360/10 *7:7cm) ;
[vertex] (v5) at (360/10 *8:7cm) ;
[vertex] (v4) at (360/10 *9:7cm) ;
[dashed] (360/10 *0 +1.5:7cm) arc  (360/10*0+1.5:360/10*1 -1.5 :7cm);
[dashed] (360/10 *1 +1.5:7cm) arc  (360/10*1+1.5:360/10*2 -1.5 :7cm);
(360/10 *2 +1.5:7cm) arc  (360/10*2+1.5:360/10*3 -1.5 :7cm);
(360/10 *3 +1.5:7cm) arc  (360/10*3+1.5:360/10*4 -1.5 :7cm);
(360/10 *4 +1.5:7cm) arc  (360/10*4+1.5:360/10*5 -1.5 :7cm);
[dashed] (360/10 *5 +1.5:7cm) arc  (360/10*5+1.5:360/10*6 -1.5 :7cm);
(360/10 *6 +1.5:7cm) arc  (360/10*6+1.5:360/10*7 -1.5 :7cm);
(360/10 *7 +1.5:7cm) arc  (360/10*7+1.5:360/10*8 -1.5 :7cm);
[dashed] (360/10 *8 +1.5:7cm) arc  (360/10*8+1.5:360/10*9 -1.5 :7cm);
[dashed] (360/10 *9 +1.5:7cm) arc  (360/10*9+1.5:360/10*10 -1.5 :7cm);

[xshift=0.2cm, yshift=0.15cm] at (360/10*0 :1.5cm) $v$;
[xshift=0.25cm] at (360/10*0 :7cm) $v_3$;
[xshift=0.25cm, yshift=0.1cm] at (360/10*1 :7cm) $v_2$;
[xshift=0.1cm, yshift=0.25cm] at (360/10 *2 :7cm) $v_1$;
[xshift=-0.1cm, yshift=0.25cm] at (360/10 *3 :7cm) $v_{10}$;
[xshift=-0.2cm, yshift=0.15cm] at (360/10 *4 :7cm) $v_9$;
[xshift=-0.25cm] at (360/10 *5:7cm) $v_8$;
[xshift=-0.25cm, yshift=-0.15cm] at (360/10 *6 :7cm) $v_7$;
[xshift=-0.1cm, yshift=-0.3cm] at (360/10 *7:7cm) $v_6$;
[xshift=0.05cm, yshift=-0.3cm] at (360/10 *8:7cm) $v_5$;
[xshift=0.2cm, yshift=-0.2cm] at (360/10 *9:7cm) $v_4$;
/→in v0/v1,v0/v2,v0/v3,v0/v4,v0/v5,v0/v6,v0/v7,v0/v8,v0/v9,v0/v10 () – (→);

[vertex] (v0) at (360/10*0 :0.1cm) ;
[vertex] (v3) at (360/10*0 :7cm) ;
[vertex] (v2) at (360/10*1 :7cm) ;
[vertex] (v1) at (360/10 *2 :7cm) ;
[vertex] (v10) at (360/10 *3 :7cm) ;
[vertex] (v9) at (360/10 *4 :7cm) ;
[vertex] (v8) at (360/10 *5:7cm) ;
[vertex] (v7) at (360/10 *6 :7cm) ;
[vertex] (v6) at (360/10 *7:7cm) ;
[vertex] (v5) at (360/10 *8:7cm) ;
[vertex] (v4) at (360/10 *9:7cm) ;
[dashed] (360/10 *0 +1.5:7cm) arc  (360/10*0+1.5:360/10*1 -1.5 :7cm);
[dashed] (360/10 *1 +1.5:7cm) arc  (360/10*1+1.5:360/10*2 -1.5 :7cm);
(360/10 *2 +1.5:7cm) arc  (360/10*2+1.5:360/10*3 -1.5 :7cm);
(360/10 *3 +1.5:7cm) arc  (360/10*3+1.5:360/10*4 -1.5 :7cm);
(360/10 *4 +1.5:7cm) arc  (360/10*4+1.5:360/10*5 -1.5 :7cm);
(360/10 *5 +1.5:7cm) arc  (360/10*5+1.5:360/10*6 -1.5 :7cm);
[dashed] (360/10 *6 +1.5:7cm) arc  (360/10*6+1.5:360/10*7 -1.5 :7cm);
[dashed] (360/10 *7 +1.5:7cm) arc  (360/10*7+1.5:360/10*8 -1.5 :7cm);
(360/10 *8 +1.5:7cm) arc  (360/10*8+1.5:360/10*9 -1.5 :7cm);
[dashed] (360/10 *9 +1.5:7cm) arc  (360/10*9+1.5:360/10*10 -1.5 :7cm);

[xshift=0.2cm, yshift=0.15cm] at (360/10*0 :1.5cm) $v$;
[xshift=0.25cm] at (360/10*0 :7cm) $v_3$;
[xshift=0.25cm, yshift=0.1cm] at (360/10*1 :7cm) $v_2$;
[xshift=0.1cm, yshift=0.25cm] at (360/10 *2 :7cm) $v_1$;
[xshift=-0.1cm, yshift=0.25cm] at (360/10 *3 :7cm) $v_{10}$;
[xshift=-0.2cm, yshift=0.15cm] at (360/10 *4 :7cm) $v_9$;
[xshift=-0.25cm] at (360/10 *5:7cm) $v_8$;
[xshift=-0.25cm, yshift=-0.15cm] at (360/10 *6 :7cm) $v_7$;
[xshift=-0.1cm, yshift=-0.3cm] at (360/10 *7:7cm) $v_6$;
[xshift=0.05cm, yshift=-0.3cm] at (360/10 *8:7cm) $v_5$;
[xshift=0.2cm, yshift=-0.2cm] at (360/10 *9:7cm) $v_4$;
/→in v0/v1,v0/v2,v0/v3,v0/v4,v0/v5,v0/v6,v0/v7,v0/v8,v0/v9,v0/v10 () – (→);

[vertex] (v0) at (360/10*0 :0.1cm) ;
[vertex] (v3) at (360/10*0 :7cm) ;
[vertex] (v2) at (360/10*1 :7cm) ;
[vertex] (v1) at (360/10 *2 :7cm) ;
[vertex] (v10) at (360/10 *3 :7cm) ;
[vertex] (v9) at (360/10 *4 :7cm) ;
[vertex] (v8) at (360/10 *5:7cm) ;
[vertex] (v7) at (360/10 *6 :7cm) ;
[vertex] (v6) at (360/10 *7:7cm) ;
[vertex] (v5) at (360/10 *8:7cm) ;
[vertex] (v4) at (360/10 *9:7cm) ;
[dashed] (360/10 *0 +1.5:7cm) arc  (360/10*0+1.5:360/10*1 -1.5 :7cm);
[dashed] (360/10 *1 +1.5:7cm) arc  (360/10*1+1.5:360/10*2 -1.5 :7cm);
(360/10 *2 +1.5:7cm) arc  (360/10*2+1.5:360/10*3 -1.5 :7cm);
(360/10 *3 +1.5:7cm) arc  (360/10*3+1.5:360/10*4 -1.5 :7cm);
(360/10 *4 +1.5:7cm) arc  (360/10*4+1.5:360/10*5 -1.5 :7cm);
[dashed] (360/10 *5 +1.5:7cm) arc  (360/10*5+1.5:360/10*6 -1.5 :7cm);
[dashed] (360/10 *6 +1.5:7cm) arc  (360/10*6+1.5:360/10*7 -1.5 :7cm);
(360/10 *7 +1.5:7cm) arc  (360/10*7+1.5:360/10*8 -1.5 :7cm);
(360/10 *8 +1.5:7cm) arc  (360/10*8+1.5:360/10*9 -1.5 :7cm);
[dashed] (360/10 *9 +1.5:7cm) arc  (360/10*9+1.5:360/10*10 -1.5 :7cm);

[xshift=0.2cm, yshift=0.15cm] at (360/10*0 :1.5cm) $v$;
[xshift=0.25cm] at (360/10*0 :7cm) $v_3$;
[xshift=0.25cm, yshift=0.1cm] at (360/10*1 :7cm) $v_2$;
[xshift=0.1cm, yshift=0.25cm] at (360/10 *2 :7cm) $v_1$;
[xshift=-0.1cm, yshift=0.25cm] at (360/10 *3 :7cm) $v_{10}$;
[xshift=-0.2cm, yshift=0.15cm] at (360/10 *4 :7cm) $v_9$;
[xshift=-0.25cm] at (360/10 *5:7cm) $v_8$;
[xshift=-0.25cm, yshift=-0.15cm] at (360/10 *6 :7cm) $v_7$;
[xshift=-0.1cm, yshift=-0.3cm] at (360/10 *7:7cm) $v_6$;
[xshift=0.05cm, yshift=-0.3cm] at (360/10 *8:7cm) $v_5$;
[xshift=0.2cm, yshift=-0.2cm] at (360/10 *9:7cm) $v_4$;
/→in v0/v1,v0/v2,v0/v3,v0/v4,v0/v5,v0/v6,v0/v7,v0/v8,v0/v9,v0/v10 () – (→);

[vertex] (v0) at (360/10*0 :0.1cm) ;
[vertex] (v3) at (360/10*0 :7cm) ;
[vertex] (v2) at (360/10*1 :7cm) ;
[vertex] (v1) at (360/10 *2 :7cm) ;
[vertex] (v10) at (360/10 *3 :7cm) ;
[vertex] (v9) at (360/10 *4 :7cm) ;
[vertex] (v8) at (360/10 *5:7cm) ;
[vertex] (v7) at (360/10 *6 :7cm) ;
[vertex] (v6) at (360/10 *7:7cm) ;
[vertex] (v5) at (360/10 *8:7cm) ;
[vertex] (v4) at (360/10 *9:7cm) ;
[dashed] (360/10 *0 +1.5:7cm) arc  (360/10*0+1.5:360/10*1 -1.5 :7cm);
[dashed] (360/10 *1 +1.5:7cm) arc  (360/10*1+1.5:360/10*2 -1.5 :7cm);
(360/10 *2 +1.5:7cm) arc  (360/10*2+1.5:360/10*3 -1.5 :7cm);
(360/10 *3 +1.5:7cm) arc  (360/10*3+1.5:360/10*4 -1.5 :7cm);
(360/10 *4 +1.5:7cm) arc  (360/10*4+1.5:360/10*5 -1.5 :7cm);
[dashed] (360/10 *5 +1.5:7cm) arc  (360/10*5+1.5:360/10*6 -1.5 :7cm);
(360/10 *6 +1.5:7cm) arc  (360/10*6+1.5:360/10*7 -1.5 :7cm);
[dashed] (360/10 *7 +1.5:7cm) arc  (360/10*7+1.5:360/10*8 -1.5 :7cm);
(360/10 *8 +1.5:7cm) arc  (360/10*8+1.5:360/10*9 -1.5 :7cm);
[dashed] (360/10 *9 +1.5:7cm) arc  (360/10*9+1.5:360/10*10 -1.5 :7cm);

[xshift=0.2cm, yshift=0.15cm] at (360/10*0 :1.5cm) $v$;
[xshift=0.25cm] at (360/10*0 :7cm) $v_3$;
[xshift=0.25cm, yshift=0.1cm] at (360/10*1 :7cm) $v_2$;
[xshift=0.1cm, yshift=0.25cm] at (360/10 *2 :7cm) $v_1$;
[xshift=-0.1cm, yshift=0.25cm] at (360/10 *3 :7cm) $v_{10}$;
[xshift=-0.2cm, yshift=0.15cm] at (360/10 *4 :7cm) $v_9$;
[xshift=-0.25cm] at (360/10 *5:7cm) $v_8$;
[xshift=-0.25cm, yshift=-0.15cm] at (360/10 *6 :7cm) $v_7$;
[xshift=-0.1cm, yshift=-0.3cm] at (360/10 *7:7cm) $v_6$;
[xshift=0.05cm, yshift=-0.3cm] at (360/10 *8:7cm) $v_5$;
[xshift=0.2cm, yshift=-0.2cm] at (360/10 *9:7cm) $v_4$;
/→in v0/v1,v0/v2,v0/v3,v0/v4,v0/v5,v0/v6,v0/v7,v0/v8,v0/v9,v0/v10 () – (→);

[vertex] (v0) at (360/10*0 :0.1cm) ;
[vertex] (v3) at (360/10*0 :7cm) ;
[vertex] (v2) at (360/10*1 :7cm) ;
[vertex] (v1) at (360/10 *2 :7cm) ;
[vertex] (v10) at (360/10 *3 :7cm) ;
[vertex] (v9) at (360/10 *4 :7cm) ;
[vertex] (v8) at (360/10 *5:7cm) ;
[vertex] (v7) at (360/10 *6 :7cm) ;
[vertex] (v6) at (360/10 *7:7cm) ;
[vertex] (v5) at (360/10 *8:7cm) ;
[vertex] (v4) at (360/10 *9:7cm) ;
[dashed] (360/10 *0 +1.5:7cm) arc  (360/10*0+1.5:360/10*1 -1.5 :7cm);
[dashed] (360/10 *1 +1.5:7cm) arc  (360/10*1+1.5:360/10*2 -1.5 :7cm);
(360/10 *2 +1.5:7cm) arc  (360/10*2+1.5:360/10*3 -1.5 :7cm);
(360/10 *3 +1.5:7cm) arc  (360/10*3+1.5:360/10*4 -1.5 :7cm);
[dashed] (360/10 *4 +1.5:7cm) arc  (360/10*4+1.5:360/10*5 -1.5 :7cm);
(360/10 *5 +1.5:7cm) arc  (360/10*5+1.5:360/10*6 -1.5 :7cm);
(360/10 *6 +1.5:7cm) arc  (360/10*6+1.5:360/10*7 -1.5 :7cm);
[dashed] (360/10 *7 +1.5:7cm) arc  (360/10*7+1.5:360/10*8 -1.5 :7cm);
(360/10 *8 +1.5:7cm) arc  (360/10*8+1.5:360/10*9 -1.5 :7cm);
[dashed] (360/10 *9 +1.5:7cm) arc  (360/10*9+1.5:360/10*10 -1.5 :7cm);

[xshift=0.2cm, yshift=0.15cm] at (360/10*0 :1.5cm) $v$;
[xshift=0.25cm] at (360/10*0 :7cm) $v_3$;
[xshift=0.25cm, yshift=0.1cm] at (360/10*1 :7cm) $v_2$;
[xshift=0.1cm, yshift=0.25cm] at (360/10 *2 :7cm) $v_1$;
[xshift=-0.1cm, yshift=0.25cm] at (360/10 *3 :7cm) $v_{10}$;
[xshift=-0.2cm, yshift=0.15cm] at (360/10 *4 :7cm) $v_9$;
[xshift=-0.25cm] at (360/10 *5:7cm) $v_8$;
[xshift=-0.25cm, yshift=-0.15cm] at (360/10 *6 :7cm) $v_7$;
[xshift=-0.1cm, yshift=-0.3cm] at (360/10 *7:7cm) $v_6$;
[xshift=0.05cm, yshift=-0.3cm] at (360/10 *8:7cm) $v_5$;
[xshift=0.2cm, yshift=-0.2cm] at (360/10 *9:7cm) $v_4$;
/→in v0/v1,v0/v2,v0/v3,v0/v4,v0/v5,v0/v6,v0/v7,v0/v8,v0/v9,v0/v10 () – (→);

[vertex] (v0) at (360/10*0 :0.1cm) ;
[vertex] (v3) at (360/10*0 :7cm) ;
[vertex] (v2) at (360/10*1 :7cm) ;
[vertex] (v1) at (360/10 *2 :7cm) ;
[vertex] (v10) at (360/10 *3 :7cm) ;
[vertex] (v9) at (360/10 *4 :7cm) ;
[vertex] (v8) at (360/10 *5:7cm) ;
[vertex] (v7) at (360/10 *6 :7cm) ;
[vertex] (v6) at (360/10 *7:7cm) ;
[vertex] (v5) at (360/10 *8:7cm) ;
[vertex] (v4) at (360/10 *9:7cm) ;
[dashed] (360/10 *0 +1.5:7cm) arc  (360/10*0+1.5:360/10*1 -1.5 :7cm);
[dashed] (360/10 *1 +1.5:7cm) arc  (360/10*1+1.5:360/10*2 -1.5 :7cm);
(360/10 *2 +1.5:7cm) arc  (360/10*2+1.5:360/10*3 -1.5 :7cm);
[dashed] (360/10 *3 +1.5:7cm) arc  (360/10*3+1.5:360/10*4 -1.5 :7cm);
(360/10 *4 +1.5:7cm) arc  (360/10*4+1.5:360/10*5 -1.5 :7cm);
(360/10 *5 +1.5:7cm) arc  (360/10*5+1.5:360/10*6 -1.5 :7cm);
(360/10 *6 +1.5:7cm) arc  (360/10*6+1.5:360/10*7 -1.5 :7cm);
[dashed] (360/10 *7 +1.5:7cm) arc  (360/10*7+1.5:360/10*8 -1.5 :7cm);
(360/10 *8 +1.5:7cm) arc  (360/10*8+1.5:360/10*9 -1.5 :7cm);
[dashed] (360/10 *9 +1.5:7cm) arc  (360/10*9+1.5:360/10*10 -1.5 :7cm);

[xshift=0.2cm, yshift=0.15cm] at (360/10*0 :1.5cm) $v$;
[xshift=0.25cm] at (360/10*0 :7cm) $v_3$;
[xshift=0.25cm, yshift=0.1cm] at (360/10*1 :7cm) $v_2$;
[xshift=0.1cm, yshift=0.25cm] at (360/10 *2 :7cm) $v_1$;
[xshift=-0.1cm, yshift=0.25cm] at (360/10 *3 :7cm) $v_{10}$;
[xshift=-0.2cm, yshift=0.15cm] at (360/10 *4 :7cm) $v_9$;
[xshift=-0.25cm] at (360/10 *5:7cm) $v_8$;
[xshift=-0.25cm, yshift=-0.15cm] at (360/10 *6 :7cm) $v_7$;
[xshift=-0.1cm, yshift=-0.3cm] at (360/10 *7:7cm) $v_6$;
[xshift=0.05cm, yshift=-0.3cm] at (360/10 *8:7cm) $v_5$;
[xshift=0.2cm, yshift=-0.2cm] at (360/10 *9:7cm) $v_4$;
/→in v0/v1,v0/v2,v0/v3,v0/v4,v0/v5,v0/v6,v0/v7,v0/v8,v0/v9,v0/v10 () – (→);

[vertex] (v0) at (360/10*0 :0.1cm) ;
[vertex] (v3) at (360/10*0 :7cm) ;
[vertex] (v2) at (360/10*1 :7cm) ;
[vertex] (v1) at (360/10 *2 :7cm) ;
[vertex] (v10) at (360/10 *3 :7cm) ;
[vertex] (v9) at (360/10 *4 :7cm) ;
[vertex] (v8) at (360/10 *5:7cm) ;
[vertex] (v7) at (360/10 *6 :7cm) ;
[vertex] (v6) at (360/10 *7:7cm) ;
[vertex] (v5) at (360/10 *8:7cm) ;
[vertex] (v4) at (360/10 *9:7cm) ;
[dashed] (360/10 *0 +1.5:7cm) arc  (360/10*0+1.5:360/10*1 -1.5 :7cm);
[dashed] (360/10 *1 +1.5:7cm) arc  (360/10*1+1.5:360/10*2 -1.5 :7cm);
(360/10 *2 +1.5:7cm) arc  (360/10*2+1.5:360/10*3 -1.5 :7cm);
(360/10 *3 +1.5:7cm) arc  (360/10*3+1.5:360/10*4 -1.5 :7cm);
[dashed] (360/10 *4 +1.5:7cm) arc  (360/10*4+1.5:360/10*5 -1.5 :7cm);
(360/10 *5 +1.5:7cm) arc  (360/10*5+1.5:360/10*6 -1.5 :7cm);
[dashed] (360/10 *6 +1.5:7cm) arc  (360/10*6+1.5:360/10*7 -1.5 :7cm);
(360/10 *7 +1.5:7cm) arc  (360/10*7+1.5:360/10*8 -1.5 :7cm);
(360/10 *8 +1.5:7cm) arc  (360/10*8+1.5:360/10*9 -1.5 :7cm);
[dashed] (360/10 *9 +1.5:7cm) arc  (360/10*9+1.5:360/10*10 -1.5 :7cm);

[xshift=0.2cm, yshift=0.15cm] at (360/10*0 :1.5cm) $v$;
[xshift=0.25cm] at (360/10*0 :7cm) $v_3$;
[xshift=0.25cm, yshift=0.1cm] at (360/10*1 :7cm) $v_2$;
[xshift=0.1cm, yshift=0.25cm] at (360/10 *2 :7cm) $v_1$;
[xshift=-0.1cm, yshift=0.25cm] at (360/10 *3 :7cm) $v_{10}$;
[xshift=-0.2cm, yshift=0.15cm] at (360/10 *4 :7cm) $v_9$;
[xshift=-0.25cm] at (360/10 *5:7cm) $v_8$;
[xshift=-0.25cm, yshift=-0.15cm] at (360/10 *6 :7cm) $v_7$;
[xshift=-0.1cm, yshift=-0.3cm] at (360/10 *7:7cm) $v_6$;
[xshift=0.05cm, yshift=-0.3cm] at (360/10 *8:7cm) $v_5$;
[xshift=0.2cm, yshift=-0.2cm] at (360/10 *9:7cm) $v_4$;
/→in v0/v1,v0/v2,v0/v3,v0/v4,v0/v5,v0/v6,v0/v7,v0/v8,v0/v9,v0/v10 () – (→);

[vertex] (v0) at (360/10*0 :0.1cm) ;
[vertex] (v3) at (360/10*0 :7cm) ;
[vertex] (v2) at (360/10*1 :7cm) ;
[vertex] (v1) at (360/10 *2 :7cm) ;
[vertex] (v10) at (360/10 *3 :7cm) ;
[vertex] (v9) at (360/10 *4 :7cm) ;
[vertex] (v8) at (360/10 *5:7cm) ;
[vertex] (v7) at (360/10 *6 :7cm) ;
[vertex] (v6) at (360/10 *7:7cm) ;
[vertex] (v5) at (360/10 *8:7cm) ;
[vertex] (v4) at (360/10 *9:7cm) ;
[dashed] (360/10 *0 +1.5:7cm) arc  (360/10*0+1.5:360/10*1 -1.5 :7cm);
[dashed] (360/10 *1 +1.5:7cm) arc  (360/10*1+1.5:360/10*2 -1.5 :7cm);
(360/10 *2 +1.5:7cm) arc  (360/10*2+1.5:360/10*3 -1.5 :7cm);
(360/10 *3 +1.5:7cm) arc  (360/10*3+1.5:360/10*4 -1.5 :7cm);
(360/10 *4 +1.5:7cm) arc  (360/10*4+1.5:360/10*5 -1.5 :7cm);
[dashed] (360/10 *5 +1.5:7cm) arc  (360/10*5+1.5:360/10*6 -1.5 :7cm);
(360/10 *6 +1.5:7cm) arc  (360/10*6+1.5:360/10*7 -1.5 :7cm);
[dashed] (360/10 *7 +1.5:7cm) arc  (360/10*7+1.5:360/10*8 -1.5 :7cm);
[dashed] (360/10 *8 +1.5:7cm) arc  (360/10*8+1.5:360/10*9 -1.5 :7cm);
(360/10 *9 +1.5:7cm) arc  (360/10*9+1.5:360/10*10 -1.5 :7cm);

[xshift=0.2cm, yshift=0.15cm] at (360/10*0 :1.5cm) $v$;
[xshift=0.25cm] at (360/10*0 :7cm) $v_3$;
[xshift=0.25cm, yshift=0.1cm] at (360/10*1 :7cm) $v_2$;
[xshift=0.1cm, yshift=0.25cm] at (360/10 *2 :7cm) $v_1$;
[xshift=-0.1cm, yshift=0.25cm] at (360/10 *3 :7cm) $v_{10}$;
[xshift=-0.2cm, yshift=0.15cm] at (360/10 *4 :7cm) $v_9$;
[xshift=-0.25cm] at (360/10 *5:7cm) $v_8$;
[xshift=-0.25cm, yshift=-0.15cm] at (360/10 *6 :7cm) $v_7$;
[xshift=-0.1cm, yshift=-0.3cm] at (360/10 *7:7cm) $v_6$;
[xshift=0.05cm, yshift=-0.3cm] at (360/10 *8:7cm) $v_5$;
[xshift=0.2cm, yshift=-0.2cm] at (360/10 *9:7cm) $v_4$;
/→in v0/v1,v0/v2,v0/v3,v0/v4,v0/v5,v0/v6,v0/v7,v0/v8,v0/v9,v0/v10 () – (→);

[vertex] (v0) at (360/10*0 :0.1cm) ;
[vertex] (v3) at (360/10*0 :7cm) ;
[vertex] (v2) at (360/10*1 :7cm) ;
[vertex] (v1) at (360/10 *2 :7cm) ;
[vertex] (v10) at (360/10 *3 :7cm) ;
[vertex] (v9) at (360/10 *4 :7cm) ;
[vertex] (v8) at (360/10 *5:7cm) ;
[vertex] (v7) at (360/10 *6 :7cm) ;
[vertex] (v6) at (360/10 *7:7cm) ;
[vertex] (v5) at (360/10 *8:7cm) ;
[vertex] (v4) at (360/10 *9:7cm) ;
[dashed] (360/10 *0 +1.5:7cm) arc  (360/10*0+1.5:360/10*1 -1.5 :7cm);
[dashed] (360/10 *1 +1.5:7cm) arc  (360/10*1+1.5:360/10*2 -1.5 :7cm);
(360/10 *2 +1.5:7cm) arc  (360/10*2+1.5:360/10*3 -1.5 :7cm);
(360/10 *3 +1.5:7cm) arc  (360/10*3+1.5:360/10*4 -1.5 :7cm);
[dashed] (360/10 *4 +1.5:7cm) arc  (360/10*4+1.5:360/10*5 -1.5 :7cm);
(360/10 *5 +1.5:7cm) arc  (360/10*5+1.5:360/10*6 -1.5 :7cm);
(360/10 *6 +1.5:7cm) arc  (360/10*6+1.5:360/10*7 -1.5 :7cm);
[dashed] (360/10 *7 +1.5:7cm) arc  (360/10*7+1.5:360/10*8 -1.5 :7cm);
[dashed] (360/10 *8 +1.5:7cm) arc  (360/10*8+1.5:360/10*9 -1.5 :7cm);
(360/10 *9 +1.5:7cm) arc  (360/10*9+1.5:360/10*10 -1.5 :7cm);

[xshift=0.2cm, yshift=0.15cm] at (360/10*0 :1.5cm) $v$;
[xshift=0.25cm] at (360/10*0 :7cm) $v_3$;
[xshift=0.25cm, yshift=0.1cm] at (360/10*1 :7cm) $v_2$;
[xshift=0.1cm, yshift=0.25cm] at (360/10 *2 :7cm) $v_1$;
[xshift=-0.1cm, yshift=0.25cm] at (360/10 *3 :7cm) $v_{10}$;
[xshift=-0.2cm, yshift=0.15cm] at (360/10 *4 :7cm) $v_9$;
[xshift=-0.25cm] at (360/10 *5:7cm) $v_8$;
[xshift=-0.25cm, yshift=-0.15cm] at (360/10 *6 :7cm) $v_7$;
[xshift=-0.1cm, yshift=-0.3cm] at (360/10 *7:7cm) $v_6$;
[xshift=0.05cm, yshift=-0.3cm] at (360/10 *8:7cm) $v_5$;
[xshift=0.2cm, yshift=-0.2cm] at (360/10 *9:7cm) $v_4$;
/→in v0/v1,v0/v2,v0/v3,v0/v4,v0/v5,v0/v6,v0/v7,v0/v8,v0/v9,v0/v10 () – (→);

[vertex] (v0) at (360/10*0 :0.1cm) ;
[vertex] (v3) at (360/10*0 :7cm) ;
[vertex] (v2) at (360/10*1 :7cm) ;
[vertex] (v1) at (360/10 *2 :7cm) ;
[vertex] (v10) at (360/10 *3 :7cm) ;
[vertex] (v9) at (360/10 *4 :7cm) ;
[vertex] (v8) at (360/10 *5:7cm) ;
[vertex] (v7) at (360/10 *6 :7cm) ;
[vertex] (v6) at (360/10 *7:7cm) ;
[vertex] (v5) at (360/10 *8:7cm) ;
[vertex] (v4) at (360/10 *9:7cm) ;
[dashed] (360/10 *0 +1.5:7cm) arc  (360/10*0+1.5:360/10*1 -1.5 :7cm);
[dashed] (360/10 *1 +1.5:7cm) arc  (360/10*1+1.5:360/10*2 -1.5 :7cm);
(360/10 *2 +1.5:7cm) arc  (360/10*2+1.5:360/10*3 -1.5 :7cm);
(360/10 *3 +1.5:7cm) arc  (360/10*3+1.5:360/10*4 -1.5 :7cm);
[dashed] (360/10 *4 +1.5:7cm) arc  (360/10*4+1.5:360/10*5 -1.5 :7cm);
(360/10 *5 +1.5:7cm) arc  (360/10*5+1.5:360/10*6 -1.5 :7cm);
[dashed] (360/10 *6 +1.5:7cm) arc  (360/10*6+1.5:360/10*7 -1.5 :7cm);
[dashed] (360/10 *7 +1.5:7cm) arc  (360/10*7+1.5:360/10*8 -1.5 :7cm);
(360/10 *8 +1.5:7cm) arc  (360/10*8+1.5:360/10*9 -1.5 :7cm);
(360/10 *9 +1.5:7cm) arc  (360/10*9+1.5:360/10*10 -1.5 :7cm);

[xshift=0.2cm, yshift=0.15cm] at (360/10*0 :1.5cm) $v$;
[xshift=0.25cm] at (360/10*0 :7cm) $v_3$;
[xshift=0.25cm, yshift=0.1cm] at (360/10*1 :7cm) $v_2$;
[xshift=0.1cm, yshift=0.25cm] at (360/10 *2 :7cm) $v_1$;
[xshift=-0.1cm, yshift=0.25cm] at (360/10 *3 :7cm) $v_{10}$;
[xshift=-0.2cm, yshift=0.15cm] at (360/10 *4 :7cm) $v_9$;
[xshift=-0.25cm] at (360/10 *5:7cm) $v_8$;
[xshift=-0.25cm, yshift=-0.15cm] at (360/10 *6 :7cm) $v_7$;
[xshift=-0.1cm, yshift=-0.3cm] at (360/10 *7:7cm) $v_6$;
[xshift=0.05cm, yshift=-0.3cm] at (360/10 *8:7cm) $v_5$;
[xshift=0.2cm, yshift=-0.2cm] at (360/10 *9:7cm) $v_4$;
/→in v0/v1,v0/v2,v0/v3,v0/v4,v0/v5,v0/v6,v0/v7,v0/v8,v0/v9,v0/v10 () – (→);

[vertex] (v0) at (360/10*0 :0.1cm) ;
[vertex] (v3) at (360/10*0 :7cm) ;
[vertex] (v2) at (360/10*1 :7cm) ;
[vertex] (v1) at (360/10 *2 :7cm) ;
[vertex] (v10) at (360/10 *3 :7cm) ;
[vertex] (v9) at (360/10 *4 :7cm) ;
[vertex] (v8) at (360/10 *5:7cm) ;
[vertex] (v7) at (360/10 *6 :7cm) ;
[vertex] (v6) at (360/10 *7:7cm) ;
[vertex] (v5) at (360/10 *8:7cm) ;
[vertex] (v4) at (360/10 *9:7cm) ;
[dashed] (360/10 *0 +1.5:7cm) arc  (360/10*0+1.5:360/10*1 -1.5 :7cm);
[dashed] (360/10 *1 +1.5:7cm) arc  (360/10*1+1.5:360/10*2 -1.5 :7cm);
(360/10 *2 +1.5:7cm) arc  (360/10*2+1.5:360/10*3 -1.5 :7cm);
(360/10 *3 +1.5:7cm) arc  (360/10*3+1.5:360/10*4 -1.5 :7cm);
[dashed] (360/10 *4 +1.5:7cm) arc  (360/10*4+1.5:360/10*5 -1.5 :7cm);
(360/10 *5 +1.5:7cm) arc  (360/10*5+1.5:360/10*6 -1.5 :7cm);
[dashed] (360/10 *6 +1.5:7cm) arc  (360/10*6+1.5:360/10*7 -1.5 :7cm);
(360/10 *7 +1.5:7cm) arc  (360/10*7+1.5:360/10*8 -1.5 :7cm);
[dashed] (360/10 *8 +1.5:7cm) arc  (360/10*8+1.5:360/10*9 -1.5 :7cm);
(360/10 *9 +1.5:7cm) arc  (360/10*9+1.5:360/10*10 -1.5 :7cm);

[xshift=0.2cm, yshift=0.15cm] at (360/10*0 :1.5cm) $v$;
[xshift=0.25cm] at (360/10*0 :7cm) $v_3$;
[xshift=0.25cm, yshift=0.1cm] at (360/10*1 :7cm) $v_2$;
[xshift=0.1cm, yshift=0.25cm] at (360/10 *2 :7cm) $v_1$;
[xshift=-0.1cm, yshift=0.25cm] at (360/10 *3 :7cm) $v_{10}$;
[xshift=-0.2cm, yshift=0.15cm] at (360/10 *4 :7cm) $v_9$;
[xshift=-0.25cm] at (360/10 *5:7cm) $v_8$;
[xshift=-0.25cm, yshift=-0.15cm] at (360/10 *6 :7cm) $v_7$;
[xshift=-0.1cm, yshift=-0.3cm] at (360/10 *7:7cm) $v_6$;
[xshift=0.05cm, yshift=-0.3cm] at (360/10 *8:7cm) $v_5$;
[xshift=0.2cm, yshift=-0.2cm] at (360/10 *9:7cm) $v_4$;
/→in v0/v1,v0/v2,v0/v3,v0/v4,v0/v5,v0/v6,v0/v7,v0/v8,v0/v9,v0/v10 () – (→);

[vertex] (v0) at (360/10*0 :0.1cm) ;
[vertex] (v3) at (360/10*0 :7cm) ;
[vertex] (v2) at (360/10*1 :7cm) ;
[vertex] (v1) at (360/10 *2 :7cm) ;
[vertex] (v10) at (360/10 *3 :7cm) ;
[vertex] (v9) at (360/10 *4 :7cm) ;
[vertex] (v8) at (360/10 *5:7cm) ;
[vertex] (v7) at (360/10 *6 :7cm) ;
[vertex] (v6) at (360/10 *7:7cm) ;
[vertex] (v5) at (360/10 *8:7cm) ;
[vertex] (v4) at (360/10 *9:7cm) ;
[dashed] (360/10 *0 +1.5:7cm) arc  (360/10*0+1.5:360/10*1 -1.5 :7cm);
[dashed] (360/10 *1 +1.5:7cm) arc  (360/10*1+1.5:360/10*2 -1.5 :7cm);
(360/10 *2 +1.5:7cm) arc  (360/10*2+1.5:360/10*3 -1.5 :7cm);
[dashed] (360/10 *3 +1.5:7cm) arc  (360/10*3+1.5:360/10*4 -1.5 :7cm);
(360/10 *4 +1.5:7cm) arc  (360/10*4+1.5:360/10*5 -1.5 :7cm);
(360/10 *5 +1.5:7cm) arc  (360/10*5+1.5:360/10*6 -1.5 :7cm);
[dashed] (360/10 *6 +1.5:7cm) arc  (360/10*6+1.5:360/10*7 -1.5 :7cm);
(360/10 *7 +1.5:7cm) arc  (360/10*7+1.5:360/10*8 -1.5 :7cm);
[dashed] (360/10 *8 +1.5:7cm) arc  (360/10*8+1.5:360/10*9 -1.5 :7cm);
(360/10 *9 +1.5:7cm) arc  (360/10*9+1.5:360/10*10 -1.5 :7cm);

[xshift=0.2cm, yshift=0.15cm] at (360/10*0 :1.5cm) $v$;
[xshift=0.25cm] at (360/10*0 :7cm) $v_3$;
[xshift=0.25cm, yshift=0.1cm] at (360/10*1 :7cm) $v_2$;
[xshift=0.1cm, yshift=0.25cm] at (360/10 *2 :7cm) $v_1$;
[xshift=-0.1cm, yshift=0.25cm] at (360/10 *3 :7cm) $v_{10}$;
[xshift=-0.2cm, yshift=0.15cm] at (360/10 *4 :7cm) $v_9$;
[xshift=-0.25cm] at (360/10 *5:7cm) $v_8$;
[xshift=-0.25cm, yshift=-0.15cm] at (360/10 *6 :7cm) $v_7$;
[xshift=-0.1cm, yshift=-0.3cm] at (360/10 *7:7cm) $v_6$;
[xshift=0.05cm, yshift=-0.3cm] at (360/10 *8:7cm) $v_5$;
[xshift=0.2cm, yshift=-0.2cm] at (360/10 *9:7cm) $v_4$;
/→in v0/v1,v0/v2,v0/v3,v0/v4,v0/v5,v0/v6,v0/v7,v0/v8,v0/v9,v0/v10 () – (→);

[vertex] (v0) at (360/10*0 :0.1cm) ;
[vertex] (v3) at (360/10*0 :7cm) ;
[vertex] (v2) at (360/10*1 :7cm) ;
[vertex] (v1) at (360/10 *2 :7cm) ;
[vertex] (v10) at (360/10 *3 :7cm) ;
[vertex] (v9) at (360/10 *4 :7cm) ;
[vertex] (v8) at (360/10 *5:7cm) ;
[vertex] (v7) at (360/10 *6 :7cm) ;
[vertex] (v6) at (360/10 *7:7cm) ;
[vertex] (v5) at (360/10 *8:7cm) ;
[vertex] (v4) at (360/10 *9:7cm) ;
(360/10 *0 +1.5:7cm) arc  (360/10*0+1.5:360/10*1 -1.5 :7cm);
[dashed] (360/10 *1 +1.5:7cm) arc  (360/10*1+1.5:360/10*2 -1.5 :7cm);
(360/10 *2 +1.5:7cm) arc  (360/10*2+1.5:360/10*3 -1.5 :7cm);
[dashed] (360/10 *3 +1.5:7cm) arc  (360/10*3+1.5:360/10*4 -1.5 :7cm);
(360/10 *4 +1.5:7cm) arc  (360/10*4+1.5:360/10*5 -1.5 :7cm);
[dashed] (360/10 *5 +1.5:7cm) arc  (360/10*5+1.5:360/10*6 -1.5 :7cm);
(360/10 *6 +1.5:7cm) arc  (360/10*6+1.5:360/10*7 -1.5 :7cm);
[dashed] (360/10 *7 +1.5:7cm) arc  (360/10*7+1.5:360/10*8 -1.5 :7cm);
(360/10 *8 +1.5:7cm) arc  (360/10*8+1.5:360/10*9 -1.5 :7cm);
[dashed] (360/10 *9 +1.5:7cm) arc  (360/10*9+1.5:360/10*10 -1.5 :7cm);

[xshift=0.2cm, yshift=0.15cm] at (360/10*0 :1.5cm) $v$;
[xshift=0.25cm] at (360/10*0 :7cm) $v_3$;
[xshift=0.25cm, yshift=0.1cm] at (360/10*1 :7cm) $v_2$;
[xshift=0.1cm, yshift=0.25cm] at (360/10 *2 :7cm) $v_1$;
[xshift=-0.1cm, yshift=0.25cm] at (360/10 *3 :7cm) $v_{10}$;
[xshift=-0.2cm, yshift=0.15cm] at (360/10 *4 :7cm) $v_9$;
[xshift=-0.25cm] at (360/10 *5:7cm) $v_8$;
[xshift=-0.25cm, yshift=-0.15cm] at (360/10 *6 :7cm) $v_7$;
[xshift=-0.1cm, yshift=-0.3cm] at (360/10 *7:7cm) $v_6$;
[xshift=0.05cm, yshift=-0.3cm] at (360/10 *8:7cm) $v_5$;
[xshift=0.2cm, yshift=-0.2cm] at (360/10 *9:7cm) $v_4$;
/→in v0/v1,v0/v2,v0/v3,v0/v4,v0/v5,v0/v6,v0/v7,v0/v8,v0/v9,v0/v10 () – (→);

Switching non-isomorphic signed $W_{10}$ with exactly five negative edges

[1]
Y. Bagheri, A.R. Moghaddamfar and F. Ramezani. The number of switching isomorphism classes of signed graphs associated with particular graphs. Discrete Applied Math., 279 (2020) 25-33.

[2]
C.L. Mallows and N.J.A. Sloane. Two-graphs, switching classes and Euler graphs are equal in number. SIAM J. Applied Math., 4 (1975) 876-880.

[3]
D. Sehrawat and B. Bhattacharjya. Signed complete graphs on six vertices and their frustration indices. Advances and Applications in Discrete Math., 24 (2020) 129-142.

[4]
P. Sole and T. Zaslavsky. A coding approach to signed graphs. SIAM J. Discrete Math., 4 (1994) 544-553.

[5]
T. Zaslavsky. Six signed Petersen graphs, and their automorphisms. Discrete Math., 312 (2012) 1558-1583.