Symmetry Parameters of Praeger-Xu Graphs

TL;DR

This paper computes the symmetry parameters, including the determining and distinguishing numbers, for Praeger-Xu graphs, revealing most are 2-distinguishable and providing the cost for this property.

Contribution

It provides the first detailed calculation of symmetry parameters for Praeger-Xu graphs, highlighting their high symmetry and 2-distinguishability.

Findings

Most Praeger-Xu graphs are 2-distinguishable

The cost of 2-distinguishing is determined for these graphs

Automorphism groups of these graphs are notably large

Abstract

Praeger-Xu graphs are connected, symmetric, 4-regular graphs that are unusual both in that their automorphism groups are large, and in that vertex stabilizer subgroups are also large. Determining number and distinguishing number are parameters that measure the symmetry of a graph by investigating additional conditions that can be imposed on a graph to eliminate its nontrivial automorphisms. In this paper, we compute the values of these parameters for Praeger-Xu graphs. Most Praeger-Xu graphs are 2-distinguishable; for these graphs we also provide the cost of 2-distinguishing.