Affiliations: FCE, University of Maribor, Smetanova 17, Maribor 2000, Slovenia. [email protected] | FME, University of Ljubljana, Aškerčeva 6, Ljubljana 1000, Slovenia. [email protected]; [email protected]
Note: [] Both authors are also researchers at Institute of mathematics, physics and mechanics. The reseseach supported in part by the Research agency of Slovenia ARRS.
Note: [] Address for correspondence: University of Ljubljana, Aškerčeva 6, Ljubljana 1000, Slovenia
Abstract: The product graph B * F of graphs B and F is an interesting model in the design of large reliable networks. Fault tolerance and transmission delay of networks are important concepts in network design. The notions are strongly related to connectivity and diameter of a graph, and have been studied by many authors. Wide diameter of a graph combines studying connectivity with the diameter of a graph. Diameter with width k of a graph G, k-diameter, is defined as the minimum integer d for which there exist at least k internally disjoint paths of length at most d between any two distinct vertices in G. Denote by $\cal{D}^W_c (G)$ the c-diameter of G and κ(G) the connectivity of G. We prove that $\cal{D}^W_{a+b}(B * F) \le r_a(F) + \cal{D}^W_b (B) + 1$ for a ≤ κ(F) and b ≤ κ(B). The Rabin number rc(G) is the minimum integer d such that there are c internally disjoint paths of length at most d from any vertex v to any set of c vertices {v1, v2, ... , vc}.
