  The $k$-truss, introduced by Cohen (2005), is a graph where every edge is
incident to at least $k$ triangles. This is a relaxation of the clique. It has
proved to be a useful tool in identifying cohesive subnetworks in a variety of
real-world graphs. Despite its simplicity and its utility, the combinatorial
and algorithmic aspects of trusses have not been thoroughly explored.
  We provide nearly-tight bounds on the edge counts of $k$-trusses. We also
give two improved algorithms for finding trusses in large-scale graphs. First,
we present a simplified and faster algorithm, based on approach discussed in
Wang & Cheng (2012). Second, we present a theoretical algorithm based on fast
matrix multiplication; this converts a triangle-generation algorithm of
Bjorklund et al. (2014) into a dynamic data structure.
