In energy-constrained wireless sensor networks (WSNs), maximizing the data collection using mobile sink(s) with minimum energy consumption is one of the practical challenging issues. In this article, we consider the problem of efficient data collection along with a pre-specified path using a mobile sink with constant speed. We refer the problem as a Maximizing data gathering with minimum energy consumption (MDGMEC) problem. So far, existing works have heuristic algorithms for MDGMEC problem. Therefore, based on network flow optimization approach, we propose a deterministic algorithm called data gathering using mobile sink for path constraint environment (DGAMSPCE) to handle the MDGMEC problem. The proposed DGAMSPCE scheme runs in polynomial time and is easily scalable for the networks with a large number of nodes. Based on the data receiving models used by the mobile sink, another algorithm called single access based data gathering using mobile sink for path constraint environment (SADGAMSPCE) is also proposed. We evaluate the proposed schemes and compare these with the existing schemes. The simulation experiments in MATLAB show that our proposed schemes outperform other existing schemes in terms of collecting the amount of data and the total energy consumption of the network, significantly.