The performance of mobile edge computing (MEC) depends critically on the quality of the wireless channels. From this viewpoint, the recently advocated intelligent reflecting surface (IRS) technique that can proactively reconfigure wireless channels is anticipated to bring unprecedented performance gain to MEC. In this paper, the problem of network throughput optimization of an IRS-assisted multi-hop MEC network is investigated, in which the phase-shifts of the IRS and the resource allocation of the relays need to be jointly optimized. However, due to the coupling among the transmission links of different hops caused by the utilization of the IRS and the complicated multi-hop network topology, it is difficult to solve the considered problem by directly applying existing optimization techniques. Fortunately, by exploiting the underlying structure of the network topology and spectral graph theory, it is shown that the network throughput can be well approximated by the second smallest eigenvalue of the network Laplacian matrix. This key finding allows us to develop an effective iterative algorithm for solving the considered problem. Numerical simulations are performed to corroborate the effectiveness of the proposed scheme.