# How to read the mathematical notation for multigraphs?

How to read the mathematical notation for multigraphs: $$E \rightarrow V \cup[V]^2$$

$$E$$ is a set of edges

$$V$$ is the set of vertices

I am having trouble especially with this part $$[V]^2$$

Source:

(Reinhard Diestel, Graph Theory 5th Edition, Springer, p.28 )

A multigraph is a pair $$(V,E)$$ of disjoint sets (of vertices and edges) together with a map $$E →V ∪ [V ]^2$$ assigning to every edge either one or two vertices, its ends. Thus, multigraphs too can have loops and multiple edges: we may think of a multigraph as a directed graph whose edge directions have been ‘forgotten’. To express that $$x$$ and $$y$$ are the ends of an edge $$e$$ we still write $$e = xy$$, though this no longer determines e uniquely.

(Reinhard Diestel, Graph Theory 5th Edition, Springer, p.28 )

• Can you give the source? Mar 20, 2020 at 19:22
• As @CyclotomicField suggests, knowing the source of this notation would be helpful. That being said, the notation makes no sense to me. Typically, $|V|$ would denote the cardinality of $V$, hence $|V|^2$ is a positive integer. However, the set $V \cup |V|^2$ is then the union of two different kinds of objects (graph vertices and integers). This doesn't make sense to me. Are you sure you copied the notation correctly? Mar 20, 2020 at 19:34
• Sorry my bad, I guess I copied the notation wrong. The font in the book (or my eyes) were bad. I guess it should be a bracket. Mar 20, 2020 at 20:20
• Based on the quoted text, $[V]^2$ is the collection $\{ \{ v_1, v_2\} : v_1, v_2 \in V\}$. This is kind of like the Cartesian product $V\times V$, but we forget the order, thus $(v_1, v_2)$ and $(v_2, v_1)$ are indistinguishable. It seems unnecessary to me to include $V$ in this union. This is meant to account for loops, but it seems to me that a loop from $v$ to itself could just as easily be denoted by $\{v,v\}$. Mar 21, 2020 at 6:04
• @MScott: Did you find your answer in the comments? Or is there still a need for further explanation? Mar 25, 2020 at 21:52