# Nomenclature and notation for some aspects of weighted directed graph.

I'm having some problem with nomenclature some structures and quantities related to weighted directed graph.

Suppose that $A \in \mathbb{R}_+^{N \times N}$ is the weighted adjacency matrix of a weighted directed graph. In this case, when a node $v$ is connected to $w$, then $a_{v,w}$ is a positive real number that indicates how strong is this connection. $a_{v,w}=0$ means that $v$ is not connected toward $w$ (but $w$ can, if $a_{w,v}>0$). In general, $a_{v,w} \neq a_{w,v}$.

I have the following sets: $$\mathcal{N}^+_v = \{w : a_{v,w}>0\}$$ $$\mathcal{N}^-_v = \{w : a_{w,v}>0\}$$ $$\mathcal{N}_v = \{w : a_{w,v}>0 \vee a_{v,w}>0\}$$ How are these set called?

Also, I have the following quantities: $$\delta^+_v = |\mathcal{N}^+_v|$$ $$\delta^-_v = |\mathcal{N}^-_v|$$ $$\delta_v = |\mathcal{N}_v|$$ $$d^+_v = \sum_{w \in \mathcal{N}^+_{v}}a_{v,w}$$ $$d^-_v = \sum_{w \in \mathcal{N}^-_{v}}a_{w,v}$$

Which are the names of the previous quantities?

On books I have found very different words and now I'm a little bit confused about the concept of "neighborhood" and of "degree".

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First consider unweighted directed graphs. I'd say that your $\mathcal{N}_v^+$ is the set of out-neighbors of $v$ and that your $\mathcal{N}_v^-$ is the set of in-neighbors. Their cardinalities are the in-degree and out-degree respectively. (Or in- and out-valency.) But if in a paper I came across $\mathcal{N}_v^+$, I would not know if it referred to in-neighbors or out-neighbors.
If I came across $\mathcal{N}_v$, I'd usually read it as the set of vertices joined by an arc to $v$, but I'd be happier if the author stated this explicitly. Its cardinality is the degree of the directed graph.
Thank you for your reply! I didn't understand what you mean when you say "But if I came across in a paper $N^+_v$, I would not know if it referred to in-neighbors or out-neighbors." – the_candyman Jun 9 '13 at 10:25