The clique-rank of a perfect graph G introduced by Fonlupt and Sebö is the linear rank of the incidence matrix of the maximum cliques of G. We study this rank for 3-chromatic perfect graphs. We prove that if, in addition, G is diamond-free, G has two distinct colorations. An immediate consequence is that the Strong Perfect Graph Conjecture holds for diamond-free graphs and for graphs with cliqu...

For F a free group of finite rank, it is shown that the fixed subgroup of any set B of endomorphisms of F has rank ≤ rank(F ), generalizing a recent result of Dicks and Ventura. The proof involves the combinatorics of derivations of groups. Some related questions are examined.

The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced triangle-free graphs with a given rank and characterize all such graphs achieving the maximum order.

We introduce Entropy Rank and Free Energy Rank, two methods to classify pages of the Web according to interest. These variants of PageRank are based on Ruelle’s thermodynamic formalism. They exhibit features from both PageRank and HITS methods.

This paper provides a complete characterization of all the rank facets of the stable set polytope STAB(G) associated with a claw-free graph G. In particular, it is shown that a claw-free graph G produces a rank facet of STAB(G) if and only if it can be obtained by means of two simple lifting procedures from three basic classes of graphs: (i) cliques, (ii) line graphs of minimal 2-connected hypo...

In this paper, we propose three new tensor decompositions for even-order tensors corresponding respectively to the rank-one decompositions of some unfolded matrices. Consequently such new decompositions lead to three new notions of (even-order) tensor ranks, to be called the M-rank, the symmetric M-rank, and the strongly symmetric M-rank in this paper. We discuss the bounds between these new te...

We study free profinite subgroups of free profinite semigroups of the same rank using, as main tools, iterated implicit operators, subword complexity and the associated entropy.