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

On the Slope of Hyperelliptic Lefschetz Fibrations and the Number of Separating Vanishing Cycles