An invariant of fiberwise Morse functions on surface bundle over $S^1$ by counting graphs

We apply Lescops construction of $mathbb{Z}$-equivariant perturbative invariant of knots and 3-manifolds to the explicit equivariant propagator of AL-paths given in arXiv:1403.8030. We obtain an invariant $hat{Z}_n$ of certain equivalence classes of fiberwise Morse functions on a 3-manifold fibered over $S^1$, which can be considered as a higher loop analogue of the Lefschetz zeta function and whose construction will be applied to that of finite type invariants of knots in such a 3-manifold. We also give a combinatorial formula for Lescops equivariant invariant $mathscr{Q}$ for 3-manifolds with $H_1=mathbb{Z}$ fibered over $S^1$. Moreover, surgery formulas of $hat{Z}_n$ and $mathscr{Q}$ for alternating sums of surgeries are given. This gives another proof of Lescops surgery formula of $mathscr{Q}$ for special kind of 3-manifolds and surgeries, which is simple in the sense that the formula is obtained easily by counting certain graphs in a 3-manifold.