3.2 Examples of coset graph

Petersen graph & dodecahedron

$N=A_5$ $H=⟨(123),(12)(45)⟩\cong S_3$ $g=(14)(25)$

$X=N\times ⟨c⟩$, $|c|=2$ $\tilde{H}=⟨(123),(12)(45)c⟩$ $g=(14)(25)$

• $\Sigma =cos(N,H,HgH)$. Note that $H$ is the stabilizer of $H\in [N:H]$. vertex = 10 valency = 3 $\Sigma$ is a Petersen graph.

• $\Gamma =cos(X,\tilde{H},\tilde{H}g\tilde{H})$ Note that $\tilde{H}$ is the stabilizer of $\tilde{H}\in \Omega$, denoted as $\omega$. Then $\tilde{H}=X_{\omega }$ and $\Gamma =cos(X,X_{\omega },X_{\omega }g{X}_{\omega })$. vertex = 20 valency = 3 $\Gamma$ is a dodecahedron.

$K_6$ & icosahedron

$N=A_5$ $H=⟨(12345),(25)(34)⟩\cong D_{10}$ $g=(23)(45)$

$X=N\times ⟨c⟩$, $|c|=2$ $\tilde{H}=⟨(12345),(25)(34)g⟩$ $g=(23)(45)$

• $\Sigma =cos(N,H,HgH)$. vertex = 6 valency = 5 $\Sigma$ is a $K_6$.

• $\Gamma =cos(X,\tilde{H},\tilde{H}g\tilde{H})$ vertex = 12 valency = 5 $\Gamma$ is a icosahedron.

$K_5$ & $K_{5[3]}$

$N=A_5$ $H=⟨(12)(34),(13)(24),(123)⟩\cong A_4$ $g=(23)(45)$

$X=N\times ⟨c⟩$, $|c|=3$ $\tilde{H}=⟨(12)(34),(13)(24),(123)c⟩$ $g=(23)(45)$

• $\Sigma =cos(N,H,HgH)$. vertex = 5 valency = 4 $\Sigma$ is a $K_5$.

• $\Gamma =cos(X,\tilde{H},\tilde{H}g\tilde{H})$ vertex = 15 valency = 12 $\Gamma$ is a $K_{5[3]}$.