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Questions related to the spectrum of graphs, defined using one of the possible variants of the discrete Laplace operator or Laplacian matrix. See https://en.wikipedia.org/wiki/Discrete_Laplace_operator
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Characterisation of walk-equivalent digraphs
This should work: $G$ is given by $A_G=\begin{bmatrix}0 & 1 &0 & 0\\0& 0 &1 &1 \\1 &0 &0 &0\\0 &0 &0 &0 \end{bmatrix}$ and $H$ given by $A_H=\begin{bmatrix}0 & 1 &0 & 0\\0& 0 &1 &0 \\0 &0 &0 &1\\1 &0 …