On Spanning Wide Diameter of Spider Web Networks

Abstract

For any bipartite graph [Formula: see text] with bipartition [Formula: see text] and [Formula: see text], a [Formula: see text]-container [Formula: see text] is a set of [Formula: see text] internally disjoint paths [Formula: see text] between two vertices [Formula: see text] and [Formula: see text] in [Formula: see text], i.e., [Formula: see text]. Moreover, if [Formula: see text] then [Formula: see text] is called a spanning [Formula: see text]-container, denoted by [Formula: see text]. The length of [Formula: see text] is [Formula: see text]. Besides, [Formula: see text] is spanning [Formula: see text]-laceable if there exists a spanning [Formula: see text]-container between any two vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. Assume that [Formula: see text] and [Formula: see text] are two distinct vertices in a spanning [Formula: see text]-laceable graph [Formula: see text]. Let [Formula: see text] be the collection of all [Formula: see text]’s. Define the spanning [Formula: see text]-wide distance between [Formula: see text] and [Formula: see text] in [Formula: see text], [Formula: see text], and the spanning [Formula: see text]-wide diameter of [Formula: see text], [Formula: see text]. In particular, the spanning wide diameter of [Formula: see text] is [Formula: see text], where [Formula: see text] is the connectivity of [Formula: see text]. In the paper we first provide the lower and upper bounds of the wide diameter of a bipartite graph, and then determine the exact values of the spanning wide diameters of the spider web networks [Formula: see text] for [Formula: see text].