# Need help with Graph notation for a subgraph

I have an undirected, unweighted, simple graph $G=(V,E)$, with $V=\left\lbrace v_1, v_2, ..., v_n \right\rbrace$ and $E= \left\lbrace e_{vu} \mid v,u \in V,\mathrm{some-condition}\right\rbrace$, where $e_{vu}=(v,u)$.

I need the formal notation that defines the set $F_{v_i}$ of all edges connecting to a given vertex $v_i$. For instance, if $V=\left\lbrace v_1, v_2, v_3, v_4\right\rbrace$ and $E=\left\lbrace e_{v_1v_2}, e_{v_1v_3}, e_{v_2v_3}, e_{v_3v_4} \right\rbrace$, then $F_{v_1}=\left\lbrace e_{v_1v_2}, e_{v_1v_3}\right\rbrace$.

I also need the notation that defines the set $W_{v_i}$ of all vertices connected to a vertex $v_i$. In the same example, $W_{v_1}=\left\lbrace v_2, v_3\right\rbrace$.

There is not commonly accepted notations for these, so you can define your own. The notations $F_{v_i}$ and $W_{v_i}$ you provided are fine: $$F_{v_i} = \{e \in E(G) \mid v_i \in e\} \quad\quad W_{v_i} = \{v \in V(G) \mid \{v,v_i\} \in E(G)\}$$