Kneser graph

Definition

Suppose $n$ and $k$ are positive integers. The Kneser graph $K{G}_{n,k}$ is an undirected graph defined as follows. Fix a set of size $n$ (we usually take the set as $\{1,2,\dots ,n\}$ for convenience). Then:

1. The vertex set of $K{G}_{n,k}$ is the collection of $k$-element subsets of the fixed set of size $n$.
2. The edge set of $K{G}_{n,k}$ is defined as follows: two vertices of $K{G}_{n,k}$ are adjacent if and only if they are disjoint when viewed as subsets of the $n$-element set.

Particular cases

Classes of cases

Condition on $n$ and $k$	Conclusion
$k>n$	The vertex set is empty, so the graph is a graph on no vertices
$k=n$	one-point graph
$n/2<k<n$	The vertex set is non-empty, but the edge set is empty, so the graph is an empty graph
$k=n/2$	In this case, the graph is a matching graph: it is a disjoint union of 2-cliques, with one 2-clique for each partition of the $n$-set into two disjoint pieces of equal size.
$1<k<n/2$	This is the interesting case. In this case, the graph is connected and non-empty, but is not a complete graph.
$k=1$	complete graph $K_n$
$k=0$	one-point graph

First few nontrivial cases

As indicated above, the interesting cases are where $1<k<n/2$. We list the first few of these:

$n$	$k$	Kneser graph $K{G}_{n,k}$
5	2	Petersen graph
6	2	Fill this in later

Odd graph case

A particular case of Kneser graphs of interest is where $n=2k+1$. This gives a one-parameter family of graphs called odd graphs. Specifically, the odd graph $O_m$ is defined as the Kneser graph with $n=2m-1$ and $k=m-1$.

Arithmetic functions

Size measures

Function	Value	Explanation
size of vertex set	$(\genfrac{}{}{0ex}{}{n}{k})=\frac{n!}{k!(n-k)!}$	By definition, it is the number of $k$-element subsets of a fixed $n$-element set.
size of edge set	$\frac{1}{2}(\genfrac{}{}{0ex}{}{n}{k,k,n-2k})=\frac{1}{2}\frac{n!}{k!k!(n-2k)!}$	This is the number of ways of dividing a set of size $n$ into two interchangeable substs of size $k$ and a left-over set of size $n-2k$.

Numerical invariants associated with vertices

We restrict attention to the case that $1<k<n/2$ for the expressions below.

Since the graph is a vertex-transitive graph, any numerical invariant associated to a vertex must be equal on all vertices of the graph. Below are listed some of these invariants:

Function	Value	Explanation
degree of a vertex	$(\genfrac{}{}{0ex}{}{n-k}{k})$	We need to determine, for a given $k$-element subset, the number of $k$-element subsets disjoint from it. This is the same as the number of $k$-element subsets of its set-theoretic complement, which has size $n-k$.
eccentricity of a vertex	$1+\mathrm{c}\mathrm{e}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}(\frac{k-1}{n-2k})$ where ceiling denotes the ceiling function: the smallest integer greater than or equal to a given number

Other numerical invariants

We restrict attention to the case $1<k<n/2$ for all the formulas below, though some of the formulas are also more generally valid.

Function	Value	Explanation
clique number	$\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{r}(n/k)$ where floor is the floor function (the greatest integer function) that returns the largest integer less than or equal to the number	For a subset of the vertex set to form a clique, the elements of the subset must be disjoint as subsets of the $n$-element set. If there are $r$ such subsets, they have a total of $kr$ elements, forcing $kr\le n$, giving the upper bound. It's also easy to see that this upper bound is achieved.
independence number	Fill this in later
chromatic number	Fill this in later
radius of a graph	$1+\mathrm{c}\mathrm{e}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}(\frac{k-1}{n-2k})$ where ceiling denotes the ceiling function: the smallest integer greater than or equal to a given number	Due to vertex-transitivity, the radius equals the eccentricity of any vertex, which has been computed above.
diameter of a graph	$1+\mathrm{c}\mathrm{e}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}(\frac{k-1}{n-2k})$ where ceiling denotes the ceiling function: the smallest integer greater than or equal to a given number	Due to vertex-transitivity, the diameter equals the eccentricity of any vertex, which has been computed above.
odd girth	$1+2\mathrm{c}\mathrm{e}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}(\frac{k}{n-2k})$ where ceiling denotes the ceiling function: the smallest integer greater than or equal to a given number
even girth	6 if $n=2k+1$ 4 if $n\ge 2k+2$