We have studied the electronic: transport properties of various topolo
gical mesoscopic rings which consist of several variable-size chains j
oined at their ends to form ideal single-channel leads. An enhancement
of the transmission coefficient is obtained when the chain lengths ar
e randomly distributed, due to the suppression of interchain interfere
nce, and, in the presence of magnetic flux, periodic magnetoconductanc
e oscillations are also shown. A single impurity changes the magneto-o
scillation pattern drastically, while for more impurities placed at ra
ndom the periodic magnetoconductance oscillations survive. Finally, th
e introduction of a transverse link between the chains destroys the br
idge-cc shape of the transmission versus energy curves obtained in the
absence of such a link.