The Strong Distance Problems

Suppose G = (V,E) is a graph and D = (V,F) is a strong orientation of G. Let u,v ∈ V, the strong distance sd(u,v) is the minimum size of strong subdigraph of D containing u and v, the strong eccentricity se(u) is the maximum strong distance sd(u,v) between u and v for all v ∈ V. The strong radius and strong diameter of D is defined as the minimum and maximum se(u) for all u ∈ V, respectively. And for any graph G, the lower (upper) orientable strong radius srad(G) (SRAD(G)) defined as the minimum (maximum) strong radius srad(D) in all possible strong oriented graph D of G, respectively; the lower (upper) orientable strong diameter sdiam(G) (SDIAM(G)) is the minimum (maximum) strong diameter sdiam(D) in all possible strong orientation D of G, respectively. This paper gives some properties for this four values. Then for the Cartesian product G×H of any two graph G and H, we find two inequalities to obtained an upper bound of srad(G×H) and sdiam(G×H). At last, we gives the exact values of srad(G) and sdiam(G) for graph G be the hypercube graph BCn, the extension hypercube graphs ECn and FCn. keywords: strong radius, strong diameter, orientable, cartesian product, hypercube graph.