Grassmann graphs are a special class of simple graphs defined from systems of subspaces. The vertices of the Grassmann graph
J
q
(
n
,
k
)
{\displaystyle J_{q}(n,k)}
k
{\displaystyle k}
n
{\displaystyle n}
vector space over a finite field of order
q
{\displaystyle q}
(
k
−
1
)
{\displaystyle (k-1)}
Many of the parameters of Grassmann graphs are
q
{\displaystyle q}
Johnson graphs , and Grassmann graphs have several of the same graph properties as Johnson graphs.
Graph-theoretic properties [ ]
J
q
(
n
,
k
)
{\displaystyle J_{q}(n,k)}
J
q
(
n
,
n
−
k
)
{\displaystyle J_{q}(n,n-k)}
For all
0
≤
d
≤
diam
(
J
q
(
n
,
k
)
)
{\displaystyle 0\leq d\leq \operatorname {diam} (J_{q}(n,k))}
d
{\displaystyle d}
(
k
−
d
)
{\displaystyle (k-d)}
ω
(
J
q
(
n
,
k
)
)
=
1
−
λ
max
λ
min
,
{\displaystyle \omega \left(J_{q}(n,k)\right)=1-{\frac {\lambda _{\max }}{\lambda _{\min }}},}
clique number of
J
q
(
n
,
k
)
{\displaystyle J_{q}(n,k)}
λ
min
{\displaystyle \lambda _{\min }}
λ
max
{\displaystyle \lambda _{\max }}
Automorphism group [ ] There is a distance-transitive subgroup of
Aut
(
J
q
(
n
,
k
)
)
{\displaystyle \operatorname {Aut} (J_{q}(n,k))}
PGL
(
n
,
q
)
{\displaystyle \operatorname {PGL} (n,q)}
In fact, unless
n
=
2
k
{\displaystyle n=2k}
k
∈
{
1
,
n
−
1
}
{\displaystyle k\in \{1,n-1\}}
Aut
(
J
q
(
n
,
k
)
)
{\displaystyle \operatorname {Aut} (J_{q}(n,k))}
≅
PGL
(
n
,
q
)
{\displaystyle \operatorname {PGL} (n,q)}
Aut
(
J
q
(
n
,
k
)
)
{\displaystyle \operatorname {Aut} (J_{q}(n,k))}
≅
PGL
(
n
,
q
)
×
C
2
{\displaystyle \operatorname {PGL} (n,q)\times C_{2}}
Aut
(
J
q
(
n
,
k
)
)
{\displaystyle \operatorname {Aut} (J_{q}(n,k))}
≅
Sym
(
[
n
]
q
)
{\displaystyle \operatorname {Sym} ([n]_{q})}
[1]
Intersection array [ ] As a consequence of being distance-transitive,
J
q
(
n
,
k
)
{\displaystyle J_{q}(n,k)}
distance-regular . Letting
d
{\displaystyle d}
J
q
(
n
,
k
)
{\displaystyle J_{q}(n,k)}
{
b
0
,
…
,
b
d
−
1
;
c
1
,
…
c
d
}
{\displaystyle \left\{b_{0},\ldots ,b_{d-1};c_{1},\ldots c_{d}\right\}}
b
j
:=
q
2
j
+
1
[
k
−
j
]
q
[
n
−
k
−
j
]
q
{\displaystyle b_{j}:=q^{2j+1}[k-j]_{q}[n-k-j]_{q}}
0
≤
j
<
d
{\displaystyle 0\leq j<d}
c
j
:=
(
[
j
]
q
)
2
{\displaystyle c_{j}:=([j]_{q})^{2}}
0
<
j
≤
d
{\displaystyle 0<j\leq d}
Spectrum [ ] The characteristic polynomial of
J
q
(
n
,
k
)
{\displaystyle J_{q}(n,k)}
φ
(
x
)
:=
∏
j
=
0
diam
(
J
q
(
n
,
k
)
)
(
x
−
(
q
j
+
1
[
k
−
j
]
q
[
n
−
k
−
j
]
q
−
[
j
]
q
)
)
(
(
n
j
)
q
−
(
n
j
−
1
)
q
)
{\displaystyle \varphi (x):=\prod \limits _{j=0}^{\operatorname {diam} (J_{q}(n,k))}\left(x-\left(q^{j+1}[k-j]_{q}[n-k-j]_{q}-[j]_{q}\right)\right)^{\left({\binom {n}{j}}_{q}-{\binom {n}{j-1}}_{q}\right)}}
[1] See also [ ] References [ ]
^ a b Brouwer, Andries E. (1989). Distance-Regular Graphs ISBN 9783642743436 OCLC 851840609 . 
![{\displaystyle {\frac {q[k]_{q}[n-k]_{q}}{2}}{\binom {n}{k}}_{q}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/453131d1cbc3c106d79353e94ed26db5af93a5ad)
![{\displaystyle \operatorname {Sym} ([n]_{q})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed4f8372d957d2599a6ea03adc43a89a82ec0462)
![{\displaystyle b_{j}:=q^{2j+1}[k-j]_{q}[n-k-j]_{q}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/160106de756c2b37438a25c749cf52207078b9f2)
![{\displaystyle c_{j}:=([j]_{q})^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60e4982914de043107f5c842183a0e53a8eb7ca8)
![{\displaystyle \varphi (x):=\prod \limits _{j=0}^{\operatorname {diam} (J_{q}(n,k))}\left(x-\left(q^{j+1}[k-j]_{q}[n-k-j]_{q}-[j]_{q}\right)\right)^{\left({\binom {n}{j}}_{q}-{\binom {n}{j-1}}_{q}\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b920a1e55f769c2b97fad0d3d11f7a5e0b24735e)