Bounds and algorithms for graph trusses Paul Burkhardt, Vance Faber, and David G. Harris Vol. 24, no. 3, pp. 191-214, 2020. Regular paper. Abstract 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 Björklund et al. (2014) into a dynamic data structure. Submitted: November 2018. Reviewed: June 2019. Revised: July 2019. Accepted: January 2020. Final: April 2020. Published: April 2020. Communicated by Xin He article (PDF) BibTeX