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We consider $M \geq 2$ pure or random quantum Ising chains of $N$ spins whenthey are coupled via a single star junction at their origins or when they arecoupled via two star junctions at the their two ends leading to the watermelongeometry. The energy gap is studied via a sequential self-dual real-spacerenormalization procedure that can be explicitly solved in terms of Kestenvariables containing the initial couplings and and the initial transversefields. In the pure case at criticality, the gap is found to decay as apower-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 thewatermelon (the case $M=2$ corresponds to $z=1$ for a single chain withperiodic boundary conditions). In the random case at criticality, the gapfollows the Infinite Disorder Fixed Point scaling $\ln \Delta_M = -N^{\psi} g$with the same activated exponent $\psi=\frac{1}{2}$ as the single chaincorresponding 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).

Star junctions and watermelons of pure or random quantum Ising chains : finite-size properties of the energy gap at criticality