We have proposed a mathematical method to investigate the matching conditions for an arterial graft in the femoral-popliteal region from a mechanical stand-point. Pulsatory blood flow, arterial wall motions, and conservation law are expressed by linear dynamical equations based on strict mechanical and constitutional considerations. To express the physiological blood flow in an actual arterial system, the tethering effects from the surrounding tissue and wall tensions were incorporated. The physiological parameters of arterial wall and tethering were utilized from reported experimental data. By complex analysis, mathematical expressions for the local impedance and reflection coefficient were obtained. They include not only blood properties such as viscosity and density, but also arterial properties including elastic modulus, radius, Poisson ratio, wall thickness, wall tension, frequency, and tethering effects from surrounding tissue. A matching condition was defined for minimizing the local impedance and reflection coefficient. The biophysical background was to reduce any mechanical mismatches, thus minimizing the disturbance of the flow velocity profile and shear stress distribution within the artery. Impedance matching in turn diminishes the negative factors for graft substitution represented by intimal hyperplasia and thrombosis. The calculated impedance and reflection coefficient inversed parabolically to functions of the resistance of the host artery, and there was one host arterial resistance that minimized the impedance and reflection coefficient. The present analysis revealed that for matching host artery with an elevated resistance, the dynamic elastic modulus of the wall of the graft that minimizes the impedance and reflection coefficient was increased. This indicates that for a host artery with a high resistance, an impedance matched stiff wall graft is preferable. For a large radius and a compliant host artery on the other hand, a large compliant graft should be linked. The present theoretical matching conditions will prevent anastomotic hyperplasia and thrombosis in graft substitution.