The average cutrank of graphs
Abstract
The cutrank of a set $X$ of vertices in a graph $G$ is defined as the rank of the $ X \times (V(G)\setminus X)$ matrix over the binary field whose $(i,j)$entry is $1$ if the vertex $i$ in $X$ is adjacent to the vertex $j$ in $V(G)\setminus X$ and $0$ otherwise. We introduce the graph parameter called the average cutrank of a graph, defined as the expected value of the cutrank of a random set of vertices. We show that this parameter does not increase when taking vertexminors of graphs and a class of graphs has bounded average cutrank if and only if it has bounded neighborhood diversity. This allows us to deduce that for each real $\alpha$, the list of inducedsubgraphminimal graphs having average cutrank larger than (or at least) $\alpha$ is finite. We further refine this by providing an upper bound on the size of obstruction and a lower bound on the number of obstructions for average cutrank at most (or smaller than) $\alpha$ for each real $\alpha\ge0$. Finally, we describe explicitly all graphs of average cutrank at most $3/2$ and determine up to $3/2$ all possible values that can be realized as the average cutrank of some graph.
 Publication:

arXiv eprints
 Pub Date:
 June 2019
 arXiv:
 arXiv:1906.02895
 Bibcode:
 2019arXiv190602895N
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 22 pages, 1 figure