Notation for graph adjacency Which is the best manner to express symbolically the adjacency relation in a graph? Is there a common notation for this?
More specifically I would like to avoid things as $\{v, w\} \in E(X)$, where $E(X)$ is "edges of the graph $X$". I've seen in some notes the notation $v \sim w$, but I don't like the idea of using a symbol like $\sim$ for a non reflexive, non transitive relation.

$v\sim w$ is the standard notation here. I'm working from home so don't have access to textbooks as sources, but I have never seen anything else used instead. From wikipedia:

The edges of a directed simple graph permitting loops $G$ is a homogeneous relation $\sim$ on the vertices of $G$ that is called the adjacency relation of $G$. Specifically, for each edge $(x,y)$, its endpoints $x$ and $y$ are said to be adjacent to one another, which is denoted $x\sim y$.

(They mention the most general case of directed graphs perhaps with loops, but the same notation is certainly used for undirected graphs.)
