Higher-Order Rank Functions on Directed Graphs

On Rank Functions of Graphs

We study rank functions (also known as graph homomorphisms onto Z), ways of imposing graded poset structures on graphs. We first look at a variation on rank functions called discrete Lipschitz functions. We relate the number of Lipschitz functions of a graph G to the number of rank functions of both G and G× E . We then find generating functions that enable us to compute the number of rank or L...

On Order and Rank of Graphs

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. Akbari, Cameron, and Khosrovshahi conjectured that the number of vertices of every reduced graph of rank r is at most m(r) = 2 − 2 if r is even and m(r) = 5 · 2(r−3)/2 − 2 if r is odd. In this article, we prove that i...

Higher-Order Low-Rank Regression

This paper proposes an efficient algorithm (HOLRR) to handle regression tasks where the outputs have a tensor structure. We formulate the regression problem as the minimization of a least square criterion under a multilinear rank constraint, a difficult non convex problem. HOLRR computes efficiently an approximate solution of this problem, with solid theoretical guarantees. A kernel extension i...

Higher rank Einstein solvmanifolds

In this paper we study the structure of standard Einstein solvmanifolds of arbitrary rank. Also the validity of a variational method for finding standard Einstein solvmanifolds is proved.

The rank-width of Directed Graphs