Resolving sets for Johnson and Kneser graphs

Abstract

A set of vertices $S$ in a graph $G$ is a {\em resolving set} for $G$ if, for any two vertices $u,v$, there exists $x\in S$ such that the distances $d(u,x) \neq d(v,x)$. In this paper, we consider the Johnson graphs $J(n,k)$ and Kneser graphs $K(n,k)$, and obtain various constructions of resolving sets for these graphs. As well as general constructions, we show that various interesting combinatorial objects can be used to obtain resolving sets in these graphs, including (for Johnson graphs) projective planes and symmetric designs, as well as (for Kneser graphs) partial geometries, Hadamard matrices, Steiner systems and toroidal grids.Comment: 23 pages, 2 figures, 1 tabl

On the Metric Dimension of Imprimitive Distance-Regular Graphs

“…It is known that strongly regular graphs with the same parameters need not have the same metric dimension: the Paley graph on 29 vertices has metric dimension 6, while the other strongly regular graphs with parameters (29,14,6,7), which fall into five switching classes, all have metric dimension 5 (see [5, Table 2]). Furthermore, the 3854 strongly regular graphs with parameters (35,16,6,8), which fall into exactly 227 switching classes [32], all have metric dimension 6 (see [5, Given what we know about the metric dimension of primitive strongly regular graphs from Theorem 1.2, we can combine this with Theorem 3.2 to obtain bounds on the metric dimension of Taylor graphs. Of course, if the descendants of a Taylor graph Γ are strongly regular graphs with logarithmic metric dimension, then this carries over to Γ.…”

On the Metric Dimension of Arithmetic Graph of a Composite Number

“…Applications of this concept exists in coin-weighing problems [4,5], Master mind game [6], digital images [7], chemistry [8], isomorphism problem [9], network discovery and verification [10]. Moreover, Bailey and Cameron used this concept and obtained bounds on the possible orders of primitive permutation groups [11] (see also [12]).…”

Search problems in vector spaces

“…On the other hand, Bailey [1] gave a semi-resolving set of size τ 2 (PG(2, q)) −1, and Héger and Takáts [12] constructed one of size 2(q + √ q) in PG(2, q), q a square prime power. Recall that A(3, q) = 2q − 1 by Theorem 1.1.…”