  We consider electronic properties of hollandite vanadate K$_2$V$_8$O$_{16}$,
a one-dimensional zigzag-chain system of $t_{2g}$ orbitals in a mixed valent
state. We first calculate the Madelung energy and obtain the relative stability
of several charge-ordering patterns to determine the most stable one that is
consistent with the observed superlattice structure. We then develop the
strong-coupling perturbation theory to derive the effective spin-orbit
Hamiltonian, starting from the triply-degenerate $t_{2g}$ orbitals in the
VO$_6$ octahedral structure. We apply an exact-diagonalization technique on
small clusters of this Hamiltonian and obtain the orbital-ordering pattern and
spin structures in the ground state. We thereby discuss the electronic and
magnetic properties of K$_2$V$_8$O$_{16}$ including predictions on the outcome
of future experimental studies.
