  In this article we find an upper and lower bound for the slope of genus g
hyperelliptic Lefschetz fibrations, which is sharp when g = 2, and demonstrate
the strong connection, in general, between the slope of hyperelliptic genus g
Lefschetz fibrations and the number of separating vanishing cycles.
Specifically, we show that the slope is greater than 4-4/g if and only if the
fibration contains separating vanishing cycles. We also improve the existing
bound on s/n, the ratio of number of separating vanishing cycles to the number
of non-separating vanishing cycles, for hyperelliptic Lefschetz fibrations of
genus g>1. In particular we show that s<=n for such fibrations when g>5.
