Isoperimetric number of a kind of Cartesian product graph

Isoperimetric number is one of the crucial parameters of interconnection networks closely related to the connectivity and parameters, such as bisection width of graphs. Azizoglu and Egecioglu obtained the isoperimetric number of Cartesian product graphs in the form of P_k × P_k × ⋯ × P_k by an embedding technique. In order to generalize such a technique, the isoperimetric number of Cartesian product graphs with form of P_l1 × P_l2 × ⋯ × P_la × C_m1 × C_m2 × ⋯ × C_mb × K_n1 × K_n2 × ⋯ × K_nc is achieved by embedding a graph S into a completed directed graph K_d (d=|V(S)| whose vertices are V(S). In the end, the relationship between the isoperimetric number of Cartesian product graphs and bisection width of graphs and Cheeger constant is discussed, and the isoperimetric number of d-dimensional k-torus T_k^d is given.