I was going through the text "Discrete Mathematics and its Applications" by Kenneth Rosen, where I found the definition of adjacent vertices in the context of directed graphs a bit different from the other books on graph theory and as well I could not quite find a clear cut reason behind the said definition.
When $(u, v)$ is an edge of the graph $G$ with directed edges, $u$ is said to be adjacent to $v$ and $v$ is said to be adjacent from $u$. The vertex $u$ is called the initial vertex of $(u, v)$, and $v$ is called the terminal or end vertex of $(u, v)$. The initial vertex and terminal vertex of a loop are the same.
Now till date the books I have read, considers(for the edge $(u,v)$) $v$ as out adjacent to $u$ and $u$ as in adjacent to $v$, which is quite logical from the direction of the edges and the concept of in-degree and out-degree goes hand in hand with it.
Now in the definition of adjacent vertices by Rosen I find it sort of gives importance to the direction of the edge and gives not much importance to the very vertex which is supposed to be our subject.
$(u,v)$ means $u \sim\sim\sim> v$
So, there is from "$u$ an edge to $v$" $->$ "$u$ is adjacent to $v$"
Similarly, there exits to "$v$ an edge from $u$" $->$ "$v$ is adjacent from $u$"
Isn't this definition a bit weird>
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