• J_q(n, k) is isomorphic to J_q(n, n – k).
• For all 0 ≤ d ≤ diam(J_q(n,k)), the intersection of any pair of vertices at distance d is (k – d)-dimensional.
• The clique number of J_q(n,k) is given by an expression in terms its least and greatest eigenvalues λ_min and λ_max:

${\displaystyle \omega \left(J_{q}(n,k)\right)=1-{\frac {\lambda _{\max }}{\lambda _{\min }}}}$ ^{[citation needed]}

There is a distance-transitive subgroup of ${\displaystyle \operatorname {Aut} (J_{q}(n,k))}$  isomorphic to the projective linear group ${\displaystyle \operatorname {P\Gamma L} (n,q)}$ .

In fact, unless ${\displaystyle n=2k}$  or ${\displaystyle k\in \{1,n-1\}}$ , ${\displaystyle \operatorname {Aut} (J_{q}(n,k))}$  ≅ ${\displaystyle \operatorname {P\Gamma L} (n,q)}$ ; otherwise ${\displaystyle \operatorname {Aut} (J_{q}(n,k))}$  ≅ ${\displaystyle \operatorname {P\Gamma L} (n,q)\times C_{2}}$  or ${\displaystyle \operatorname {Aut} (J_{q}(n,k))}$  ≅ ${\displaystyle \operatorname {Sym} ([n]_{q})}$  respectively.^[1]

As a consequence of being distance-transitive, ${\displaystyle J_{q}(n,k)}$  is also distance-regular. Letting ${\displaystyle d}$  denote its diameter, the intersection array of ${\displaystyle J_{q}(n,k)}$  is given by ${\displaystyle \left\{b_{0},\ldots ,b_{d-1};c_{1},\ldots c_{d}\right\}}$  where:

• ${\displaystyle b_{j}:=q^{2j+1}[k-j]_{q}[n-k-j]_{q}}$  for all ${\displaystyle 0\leq j<d}$ .
• ${\displaystyle c_{j}:=([j]_{q})^{2}}$  for all ${\displaystyle 0<j\leq d}$ .

• The characteristic polynomial of ${\displaystyle J_{q}(n,k)}$  is given by

${\displaystyle \varphi (x):=\prod \limits _{j=0}^{\operatorname {diam} (J_{q}(n,k))}\left(x-\left(q^{j+1}[k-j]_{q}[n-k-j]_{q}-[j]_{q}\right)\right)^{\left({\binom {n}{j}}_{q}-{\binom {n}{j-1}}_{q}\right)}}$ .^[1]

• Grassmannian
• Johnson graph