In the previous work (Jafarizadeh and Sufiani 2008 Phys. Rev. A 77 022315), by using some techniques such as stratification and spectral distribution associated with the graphs, perfect state transfer (PST) of a qubit (spin 1/2 particle) over distance-regular spin networks was discussed. In this paper, optimal transfer of an arbitrary d-level quantum state (qudit) over antipodes of more general networks called pseudo-distance-regular networks, is investigated. In other words, by using the same spectral analysis techniques and algebraic structures of pseudo-distance-regular graphs, we give an explicit analytical formula for suitable coupling constants in the specific Hamiltonians so that the state of a particular qudit initially encoded on one site will optimally evolve into the opposite site without any dynamical control, i.e., we show how to analytically derive the parameters of the system so that optimal state transfer can be achieved. Also, for the specific form of Hamiltonians that we consider, necessary conditions in order for PST to be achieved are given. Finally, for these Hamiltonians, PST and optimal imperfect ST over some important examples of pseudo-distance regular networks are discussed.