  We consider $M \geq 2$ pure or random quantum Ising chains of $N$ spins when
they are coupled via a single star junction at their origins or when they are
coupled via two star junctions at the their two ends leading to the watermelon
geometry. The energy gap is studied via a sequential self-dual real-space
renormalization procedure that can be explicitly solved in terms of Kesten
variables containing the initial couplings and and the initial transverse
fields. In the pure case at criticality, the gap is found to decay as a
power-law $\Delta_M \propto N^{-z(M)} $ with the dynamical exponent
$z(M)=\frac{M}{2}$ for the single star junction (the case $M=2$ corresponds to
$z=1$ for a single chain with free boundary conditions) and $z(M)=M-1$ for the
watermelon (the case $M=2$ corresponds to $z=1$ for a single chain with
periodic boundary conditions). In the random case at criticality, the gap
follows the Infinite Disorder Fixed Point scaling $\ln \Delta_M = -N^{\psi} g$
with the same activated exponent $\psi=\frac{1}{2}$ as the single chain
corresponding to $M=2$, and where $g$ is an $O(1)$ random positive variable,
whose distribution depends upon the number $M$ of chains and upon the geometry
(star or watermelon).
