  The present study shows that the electronic structure of the
$\beta$-$AV\_6O\_{15}$ family of compounds ($A = Sr, Ca, Na ...$) is based on
weakly interacting two-leg ladders, in contrast with the zig-zag chain model
one could expect from their crystal structure. Spin dimer analysis, based on
extended H\"{u}ckel tight-binding calculations, was performed to determine the
structure of the dominant transfer and magnetic interactions in the room
temperature $\beta$-$SrV\_6O\_{15}$ phase. Two different two-legs ladders,
associated with different charge/spin orders are proposed to describe these
one-dimensional $\beta$-type systems. The antiferromagnetic ladders are packed
in an 'IPN' geometry and coupled to each other through weak antiferromagnetic
interactions. This arrangement of the dominant interactions explains the
otherwise surprising similarities of the optical conductivity and Raman spectra
for the one-dimensional $\beta$-type phases and the two-dimensional
$\alpha$-type ones such as the well-known $\alpha^\prime$-$NaV\_2O\_5$ system.
