Diameters in graphs

Diameters of Cayley Graphs

We show that for integers k ≥ 2 and n ≥ 3, the diameter of the Cayley graph of SLn(Z/kZ) associated to a standard two-element generating set, is at most a constant times n 2 ln k. This answers a question of A. Lubotzky concerning SLn(Fp) and is unexpected because these Cayley graphs do not form an expander family. Our proof amounts to a quick algorithm for finding short words representing eleme...

Graphs Over Time: Densification and Shrinking Diameters

How do real graphs evolve over time? What are “normal” growth patterns in social, technological, and information networks? Many studies have discovered patterns in static graphs, identifying properties in a single snapshot of a large network, or in a very small number of snapshots; these include heavy tails for inand out-degree distributions, communities, small-world phenomena, and others. Howe...

Diameters of Degree Graphs of Nonsolvable Groups

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 ...

Diameters of Cayley graphs of Chevalley groups

We show that for integers k ≥ 2 and n ≥ 3, the diameter of the Cayley graph of SLn(Z/kZ) associated to a standard two-element generating set, is at most a constant times n ln k. This answers a question of A. Lubotzky concerning SLn(Fp) and is unexpected because these Cayley graphs do not form an expander family. Our proof amounts to a quick algorithm for finding short words representing element...