On Spanning Wide Diameter of Spider Web Networks.

For any bipartite graph G with bipartition V 1 and V 2 , a t -container C t (u , v) is a set of t internally disjoint paths P 1 , P 2 , ... , P t between two vertices u ∈ V 1 and v ∈ V 2 in G , i.e., C t (u , v) = { P 1 , P 2 , ... , P t }. Moreover, if V (P 1) ∪ V (P 2) ∪ ⋯ ∪ V (P t) = V (G) then C t (u , v) is called a spanning t -container, denoted by C t s l (u , v). The length of C t s l (u , v) = { P 1 , P 2 , ... , P t } is l (C t s l (u , v)) = max { l (P i) | 1 ≤ i ≤ t }. Besides, G is spanning t -laceable if there exists a spanning t -container between any two vertices u ∈ V 1 and v ∈ V 2 in G. Assume that u ∈ V 1 and v ∈ V 2 are two distinct vertices in a spanning t -laceable graph G. Let D t s l (u , v) be the collection of all C t s l (u , v) 's. Define the spanning t -wide distance between u and v in G , d t s l (u , v) = min { l (C t s l (u , v)) | C t s l (u , v) ∈ D t s l (u , v) } , and the spanning t -wide diameter of G , D t s l (G) = max { d t s l (u , v) | u ∈ V 1 , v ∈ V 2 }. In particular, the spanning wide diameter of G is D κ s l (G) , where κ is the connectivity of G. 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 S W (m , n) for n = 2 , 4. [ABSTRACT FROM AUTHOR]