Some resolving parameters in a class of Cayley graphs

TL;DR

This paper constructs a specific class of Cayley graphs called Toeplitz graphs, analyzes their properties, and determines key resolving parameters like minimal resolving sets and strong metric dimension, highlighting their combinatorial significance.

Contribution

It introduces a new class of Cayley graphs, analyzes their symmetry and spectral properties, and computes their resolving parameters, which were previously unexplored for this class.

Findings

The graphs are vertex transitive but not edge transitive.

They are not distance regular.

Resolved minimal resolving set, doubly resolving set, and strong metric dimension for the class.

Abstract

Resolving parameters is a fundamental area of combinatorics with applications not only to many branches of combinatorics but also to other sciences. In this article, we construct a class of Toeplitz graphs, and will be denoted by $T_{2n}(W)$, so that they are Cayley graphs. First, we review some of the features of this class of graphs. In fact, this class of graphs are vertex transitive, and by calculating the spectrum of the adjacency matrix related with them, we show that this class of graphs cannot be edge transitive. Moreover, we show that this class of graphs cannot be distance regular, and since the computing resolving parameters of a class of graphs such that are not distance regular is more difficult, then we regard this as justification for our focus on some resolving parameters. In particular, we determine the minimal resolving set, doubly resolving set and strong metric dimension for this class of graphs.