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removed note at the end, and added a note at the beginning on why on the answer is kept
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Cyriac Antony
Cyriac Antony
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(This characterisation is already given here; It is added as cw-answer for better visibility).

Quote from the paper Jørgensen, Normally Regular Digraphs (2014)

(The $(i, j)$ entry of $AA^t$ (respectively $A^tA$) is the number of common out-neighbours (respectively in-neighbours) of $x_i$ and $x_j$).

We say that a digraph is normal if its adjacency matrix $A$ is normal, i.e., if $AA^t = A^tA$. It follows that a digraph is normal if and only if for any two (not necessarily distinct) vertices $x$ and $y$, the number of common out-neighbours of $x$ and $y$ is equal to the number of common in-neighbours of $x$ and $y$.

This seems to answer you question. Please let me know if this characterization is wrong.

Quote from the paper Jørgensen, Normally Regular Digraphs (2014)

(The $(i, j)$ entry of $AA^t$ (respectively $A^tA$) is the number of common out-neighbours (respectively in-neighbours) of $x_i$ and $x_j$).

We say that a digraph is normal if its adjacency matrix $A$ is normal, i.e., if $AA^t = A^tA$. It follows that a digraph is normal if and only if for any two (not necessarily distinct) vertices $x$ and $y$, the number of common out-neighbours of $x$ and $y$ is equal to the number of common in-neighbours of $x$ and $y$.

This seems to answer you question. Please let me know if this characterization is wrong.

(This characterisation is already given here; It is added as cw-answer for better visibility).

Quote from the paper Jørgensen, Normally Regular Digraphs (2014)

(The $(i, j)$ entry of $AA^t$ (respectively $A^tA$) is the number of common out-neighbours (respectively in-neighbours) of $x_i$ and $x_j$).

We say that a digraph is normal if its adjacency matrix $A$ is normal, i.e., if $AA^t = A^tA$. It follows that a digraph is normal if and only if for any two (not necessarily distinct) vertices $x$ and $y$, the number of common out-neighbours of $x$ and $y$ is equal to the number of common in-neighbours of $x$ and $y$.

Post Made Community Wiki by Cyriac Antony
added short explanation given in paper
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Cyriac Antony
Cyriac Antony
  • 191
  • 8

Quote from the paper Jørgensen, Normally Regular Digraphs (2014)

(The $(i, j)$ entry of $AA^t$ (respectively $A^tA$) is the number of common out-neighbours (respectively in-neighbours) of $x_i$ and $x_j$).

We say that a digraph is normal if its adjacency matrix $A$ is normal, i.e., if $AA^t = A^tA$. It follows that a digraph is normal if and only if for any two (not necessarily distinct) vertices $x$ and $y$, the number of common out-neighbours of $x$ and $y$ is equal to the number of common in-neighbours of $x$ and $y$.

This seems to answer you question. Please let me know if this characterization is wrong.

Quote from the paper Jørgensen, Normally Regular Digraphs (2014)

We say that a digraph is normal if its adjacency matrix $A$ is normal, i.e., if $AA^t = A^tA$. It follows that a digraph is normal if and only if for any two (not necessarily distinct) vertices $x$ and $y$, the number of common out-neighbours of $x$ and $y$ is equal to the number of common in-neighbours of $x$ and $y$.

This seems to answer you question. Please let me know if this characterization is wrong.

Quote from the paper Jørgensen, Normally Regular Digraphs (2014)

(The $(i, j)$ entry of $AA^t$ (respectively $A^tA$) is the number of common out-neighbours (respectively in-neighbours) of $x_i$ and $x_j$).

We say that a digraph is normal if its adjacency matrix $A$ is normal, i.e., if $AA^t = A^tA$. It follows that a digraph is normal if and only if for any two (not necessarily distinct) vertices $x$ and $y$, the number of common out-neighbours of $x$ and $y$ is equal to the number of common in-neighbours of $x$ and $y$.

This seems to answer you question. Please let me know if this characterization is wrong.

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Cyriac Antony
Cyriac Antony
  • 191
  • 8

Quote from the paper Jørgensen, Normally Regular Digraphs (2014)

We say that a digraph is normal if its adjacency matrix $A$ is normal, i.e., if $AA^t = A^tA$. It follows that a digraph is normal if and only if for any two (not necessarily distinct) vertices $x$ and $y$, the number of common out-neighbours of $x$ and $y$ is equal to the number of common in-neighbours of $x$ and $y$.

This seems to answer you question. Please let me know if this characterization is wrong.