Wide Diameters of (n, k)-Star Graphs

An (n, k)-star graph (Sn,k) refers to a generalized version of an n-star graph (Sn), where Sn,n−1 and Sn are isomorphic and Sn,1 is obviously a complete graph of n vertices (Kn). Since Lin et al. (2004) already calculated the wide diameters in (n, n−1)-star and (n, 1)-star graphs, this study only considers an Sn,k with 2≤k≤n−2. Lin et al. (2008) also computed upper and lower bounds of the wide diameter of an Sn,k. However, the upper bound is larger than the lower bound by one if ⎣n/2⎦+1≤k≤n−2. Therefore, this work shortens the longest paths in the longest containers constructed by Lin et al. for ⎣n/2⎦+1≤k≤n−2 and n≥7, such that the upper bound can be reduced at least one. Moreover, a better lower bound is provided for ⎣n/2⎦+1≤k≤n−2 and n<7. The wide diameter of an Sn,k can then be determined since its upper and lower bounds are the same, and it is its diameter plus 1 for ⎣n/2⎦+1≤k≤n−2 and n≥7, or plus 2 for otherwise.