Notation in graph theory I've run across a notation that doesn't make much sense to me after reading through it in Douglas West's "Introduction to Graph Theory". The notation is the following:

$G=\frac{n}{2m} K_{2m/n}$ where 2m is a multiple of n

This appears in Example 6.3.17 (in the second edition), which is supposed to show that the bound of the crossing number inequality is asymptotically best possible.
At first I thought this notation meant $\frac{n}{2m}$ disjoint copies of $K_{2m/n}$ but since $2m$ is a multiple of $n$ this doesn't make sense unless $2m=n$.
Also, the total number of vertices is $n$ which if I calculate based on my initial thought turns out to be $\frac{n}{2m}\frac{m}{2n}=1$.
Any idea what this notation means?
 A: From reading the proof that follows, we can see that your interpretation is correct, but the parameters are off.
It should be $G = \frac{n^2}{2m} K_{2m/n}$, where the constraint is that $\frac{n^2}{2m}$ and $\frac{2m}{n}$ should both be integers. (This is achieved by taking an even $n$, picking $d$ to be any divisor of $n$, and letting $m = \frac12 nd$.) It's a slightly awkward parametrization, but the point was to get a graph $G$ with $n$ vertices and approximately $m$ edges. This works out:

*

*With $\frac{n^2}{2m}$ copies of $K_{2m/n}$, we get $\frac{n^2}{2m} \cdot \frac{2m}{n} = n$ vertices.

*With $\frac{n^2}{2m}$ copies of $K_{2m/n}$, each copy has $\binom{2m/n}{2} \approx \frac{2m^2}{n^2}$ edges, so taking $\frac{n^2}{2m}$ copies gives us approximately $m$ edges. (The exact number of edges is $m - \frac n2$: when $m \ge 4n$, this is within a constant factor of $m$.)

Finally, since the crossing number of $K_{2m/n}$ is at least $\frac1{64}(\frac{2m}{n})^4 = \frac{m^4}{4n^4}$, the crossing number of $\frac{n^2}{2m}$ copies of $K_{2m/n}$ is at least $\frac{n^2}{2m} \cdot \frac{m^4}{4n^4} = \frac{m^3}{8n^2}$, which is also what we get at the end of this example.

It should be noted that this is listed in Douglas West's errata for the textbook: the section "Corrections for Chapters 1-7 IMPLEMENTED for the second printing" contains

p265 - Exm6.3.17: In the definition of the example graph $G$, the number of copies of the complete graph should be $n^2/(2m)$

