Quantum transport effects in a molecular junction composed of a nanosized graphite sheet and two gold leads are studied on the basis of Landauer's formalism. The formulation for tunneling current by Caroli, Combescot, Nozieres, and Saint-James is extended to incorporate multiple interactions in a metal-insulator connecting region. A large variation of conductance is obtained, depending on the manner of connections between a graphite sheet and two gold chains. Connections between zigzag-edge sites and gold chains have significant transport effects. Graphite sheets of several sizes are studied to increase our understanding of the exponential law of conductance g = g(0)e(-gammaL), in which L is the molecular length and gamma is the damping factor. Reverse exponential law with negative gamma is observed in graphite sheets with zigzag edges. That is, the conductance is enhanced with an increase in L. This interesting behavior of conductance is due to the unique nature of the HOMO and LUMO localized in the zigzag-edge regions and the remarkable decrease in the HOMO-LUMO gap with L. Quantum transport effects in graphite sheets with defects such as a disordered zigzag edge are also studied. It is found that the regular zigzag-edge structures lead to effective quantum transport in graphite sheets.