Example graphs

The path graph \(P_n\)

A path graph is a graph whose \(n\) vertices can be listed in the order \(0, 1, \ldots , n-1\) such that the edges are \((i, i+1)\) for \(i = 0, 1 \ldots , n-2\). For example, the undirected graphs \(P_3\) and \(P_6\) are shown below.

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for n in [3,6]:
  G=graphs.PathGraph(n)
  G.show(figsize=4,graph_border=True,title=f'P{n}')

The cycle graph \(C_n\)

The cycle graph \(C_n\) is a graph whose $n$ vertices can be listed in the order \(0, 1, \ldots , n\) such that the edges are \((i, i+1)\) for \(i = 0, 1, \ldots , n-2\), and also \((0,n-1)\). For example, \(C_3\), \(C_5\), and \(C_{12}\) are

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for n in [3,5,12]:
    graphs.CycleGraph(n).show(figsize=3,title='C%s:' %(n))

The complete graph \(K_n\)

The complete graph \(K_n\) is a graph with \(n\) vertices and \(|E(K_n)|=\binom{n}{2}\) edges.
For example, \(K_2\), \(K_4\), and \(K_7\) are

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for n in [2,4,7]:
    graphs.CompleteGraph(n).show(figsize=3,title='K%s:' %(n))

The hypercube graph \(Q_n\)

The hypercube graph \(Q_n\) has \(|V(Q_n)|=2^n\) vertices and \(|E(Q_n)|=2^{n-1}n\) edges. For example, \(Q_2\) and \(Q_5\) are

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for n in [2,5,9]:
    if n>3:
        graphs.CubeGraph(n).show(figsize=3,vertex_size=20,vertex_labels=false,title='Q%s:' %(n))
    else:
        graphs.CubeGraph(n).show(figsize=3,title='Q%s:' %(n))

Here is a 3D image of \(Q_4\):

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n=4
graphs.CubeGraph(n).show3d(title='Q%s:' %(n))

Here is a 3D image of \(Q_3\) and \(Q_4\):

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for n in [3,4]:
    graphs.CubeGraph(n).show3d(title='Q%s:' %(n))