$12\mathrm{.}2.q-$dimensional torus networks

This problem deals with $m-$ary $q-$cubes, i$.$e$.,q-$dimensional torus networks with sides of length

$m.$

a$.$ Show that an $m-$ary $q-$cube is node$-$symmetric in the sense that the network looks exactly

the same when viewed from any of its nodes.

b$.$ Show that the sum of distances from any node of a $2$D torus to all other nodes is $m\frac{m^2-1}{2}$

if $m$ is odd and $\frac{m^3}{2}$ if $m$ is even. These lead to simple closed$-$form expressions for the

average internode distance in the two cases.

c$.$ Show that the generalized forms for the expressions of part $($b$)$ in the case of an $m-$ary

$q-$cube are $q(m^2-1)\frac{m^{q-1}}{4}$ and $q{m}^{q+\frac{1}{4}},$ respectively.