Relationships between symmetry-based graph measures

Abstract

This paper addresses the problem of comparing different measures of graph symmetry. Two measures, each based on the number and respective sizes of the vertex orbits of the automorphism group or a graph, are compared. A real valued distance measure is used to compare the symmetry measures by establishing the limiting value of the distances for several well known classes of graphs.

Introduction

A number of different symmetry measures for networks/graphs have been developed and analyzed, see [1], [2], [3], [4], [5]. The differences are due in part to the fact that symmetry can be interpreted in different ways, e.g. by means of knot theory [6] or the automorphism group of a graph. Here, we investigate symmetry in graphs in relation to the automorphism group, with special emphasis on the vertex orbits of the group. One problem we face in this investigation is that there is no general formula for the size of the automorphism group of a graph [7], [8], although many special cases are known [9]. Symmetry based on vertex orbits has long been used to define measures of the structural complexity of graphs. For example, see the seminal work due to Mowshowitz [5], [10], [11], [12] who analyzed several variants of these measures representing the structural information content of a graph.

Symmetry measures have been applied in many disciplines. Such measures have been used in structural chemistry and chemotherapies for characterizing molecular graphs numerically and to solve QSAR/QSPR problems, see [1], [2], [13]. MacArthur et al. [14] determined the size and structure of the automorphism groups of real networks, and discussed how symmetry can be used for applications. A similar study has been performed by Ball and Geyer-Schulz [15] who analyzed symmetries in large graphs. Finally, the role of symmetry in network aesthetics has been investigated by Chen et al. [16]. Finally Algebraic Graph Theory is a classical field where symmetry has been investigated extensively, see [17], [18], [19].

As indicated above, many graph measures have been defined [20], [21], [13]. However, relatively few of them have been examined in much depth. This also holds for symmetry-based graph measures defined in terms of graph automorphisms [3], [5]. This paper focuses on the relationships between symmetry measures for graphs, utilizing real valued distance measures. More precisely, we study limiting values of distances $d(I_1,I_2)$ between measures (see Section (2.2)) as the number of vertices goes to infinity Here, $I_1:\mathcal{G}\to {\mathbb{R}}_{+}$ and $I_2:\mathcal{G}\to {\mathbb{R}}_{+}$ are two graph measures and $\mathcal{G}$ is a class of graphs. In particular, we investigate the symmetry measures $\delta$ [3] and $I_a$ [5].

One elegant way to infer symmetry measures for graphs is based on using certain graph polynomials. In general, there exist various ways and techniques to define graph polynomials to determine combinatorial information and algebraic properties from a graph, see [22]. For instance, graph polynomials [23], [22] have been used as counting polynomials and to define graph measures [3], [24]. In [3], Dehmer et al. introduced the concept of the so-called orbit polynomial denoted by $O_G(z)$, which is defined in terms of the sizes of vertex orbits and their respective multiplicities. The unique, positive root $\delta ⩽1$ of the modified polynomial $1-O_G(z)$ has been shown to serve as a measure of symmetry of a graph, see [3]. The aim of this paper is to establish limiting values of $d(\delta ,I_a)$ for some special graph classes that have proven useful in chemistry and related disciplines.