On the number of k-dominating independent sets

We study the existence and the number of $k$-dominating independent sets in certain graph families. While the case $k=1$ namely the case of maximal independent sets - which is originated from Erd\H{o}s and Moser - is widely investigated, much less is known in general. In this paper we settle the question for trees and prove that the maximum number of $k$-dominating independent sets in $n$-vertex graphs is between $c_k\cdot\sqrt[2k]{2}^n$ and $c_k'\cdot\sqrt[k+1]{2}^n$ if $k\geq 2$, moreover the maximum number of $2$-dominating independent sets in $n$-vertex graphs is between $c\cdot 1.22^n$ and $c'\cdot1.246^n$. Graph constructions containing a large number of $k$-dominating independent sets are coming from product graphs, complete bipartite graphs and with finite geometries. The product graph construction is associated with the number of certain MDS codes.Comment: 13 page

On the adjacency algebras of near hexagons with an order

Suppose is a finite near hexagon of order (s, t) having v points. For every , let denote the adjacency matrix of the graph defined on the points by the distance i relation. We perform a study of the real algebra generated by the 's, and take a closer look to the structure of these algebras for all known examples of . Among other things, we show that a certain number (which is a function of s, t and v) must be integral. This allows us to exclude certain near hexagons whose (non)existence was already open for about 15 years. In the special case , we also show that the embedding rank of the near hexagon is at least the number , and that the near hexagon has non-full projective dimensions with vector dimension equal to d(S)

On the Number of k-Dominating Independent Sets

We study the existence and the number of k-dominating independent sets in certain graph families. While the case k=1 namely the case of maximal independent sets-which is originated from Erdos and Moser-is widely investigated, much less is known in general. In this paper we settle the question for trees and prove that the maximum number of k-dominating independent sets in n-vertex graphs is between ck·22kn and ck'·2k+1n if k≥2, moreover the maximum number of 2-dominating independent sets in n-vertex graphs is between c·1.22n and c'·1.246n. Graph constructions containing a large number of k-dominating independent sets are coming from product graphs, complete bipartite graphs, and finite geometries. The product graph construction is associated with the number of certain Maximum Distance Separable (MDS) codes. © 2016 Wiley Periodicals, Inc