Circulant example -- Sage

G = graphs.CirculantGraph(9,[1,3]) G

show(G.adjacency_matrix())

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrrr} 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 \end{array}\right)$

M = Matrix(ZZ, [[1, 1, 0, 5, 0, 0, -2, 0, 1], [0, 6, 0, 0, 3, 0, 2, 1, 4], [0, 0, 2, 0, 2, 1, 1, 0, 4], [0, 5, 1, 1, 4, 0, 2, 0, 5]]) show(M)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrrr} 1 & 1 & 0 & 5 & 0 & 0 & -2 & 0 & 1 \\ 0 & 6 & 0 & 0 & 3 & 0 & 2 & 1 & 4 \\ 0 & 0 & 2 & 0 & 2 & 1 & 1 & 0 & 4 \\ 0 & 5 & 1 & 1 & 4 & 0 & 2 & 0 & 5 \end{array}\right)$

D = diagonal_matrix([1, 1, 1, -1]) show(D)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{array}\right)$

A = M.transpose() * D * M show(A)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrrr} 1 & 1 & 0 & 5 & 0 & 0 & -2 & 0 & 1 \\ 1 & 12 & -5 & 0 & -2 & 0 & 0 & 6 & 0 \\ 0 & -5 & 3 & -1 & 0 & 2 & 0 & 0 & 3 \\ 5 & 0 & -1 & 24 & -4 & 0 & -12 & 0 & 0 \\ 0 & -2 & 0 & -4 & -3 & 2 & 0 & 3 & 0 \\ 0 & 0 & 2 & 0 & 2 & 1 & 1 & 0 & 4 \\ -2 & 0 & 0 & -12 & 0 & 1 & 5 & 2 & 0 \\ 0 & 6 & 0 & 0 & 3 & 0 & 2 & 1 & 4 \\ 1 & 0 & 3 & 0 & 0 & 4 & 0 & 4 & 8 \end{array}\right)$

G.is_isomorphic(Graph(A - diagonal_matrix(A.diagonal())))

True

True

Circulant example

1430 days ago by hthall