Relations and bounds for the zeros of graph polynomials using vertex orbits
Introduction
A number of structural graph measures designed to capture symmetry have been developed [1], [2], [3], [4], [5]. Most of these measures depend on determining the orbits of the automorphism group of a graph, a task that can be computationally demanding, especially if it is first necessary to compute the automorphism group itself. There is no general formula for the size of the automorphism group of a graph [6], [7], and no efficient algorithm for determining the elements of the group. Symmetry is an indicator of structural complexity that can be useful for analyzing graphs. For example, symmetry-based measures have been used to characterize network aesthetics [8]. These measures have also been used to determine the structural complexity of molecules represented by graphs [5]. A shortcoming of symmetry-based measures is the so-called problem of degeneracy [9]. This problem is characterized by the existence of many pairs of non-isomorphic graphs with the same measured value of the index. An attempt to overcome this problem in the case of classical orbit-based entropy measures is presented in [10].
In this paper, we elaborate further on the unique, positive root of the orbit polynomial introduced in [1]. This polynomial is designed as an aid in detecting symmetry in networks. In general, graph polynomials have proven useful in many disciplines such as mathematical chemistry, bioinformatics, and applied mathematics, see [11], [12], [13], [14]. In these areas, graph polynomials have been used for counting [13] and also for defining new topological graph measures, see [1], [15]. Moreover, well-known graph polynomials such as the chromatic, Tutte, and Jones polynomials have turned out to be useful and important graph invariants for studying problems in applied and pure graph theory, see [16], [17].
This paper extends results in [1]. In that earlier paper, we introduced the orbit polynomial OG(z) and the related graph polynomial . The root δ of can be calculated in a straightforward way using a standard root finding method, see [18]. Various properties of δ have been established in [1]. This paper extends the earlier results by proving bounds on δ for graphs with given orbit sizes and multiplicities. Analytical results are given for graphs with two and three orbit sizes and also for more general graphs. In particular, properties of δ are presented for exhaustive sets of isomers of hydrocarbons with 14 carbon atoms. The methods presented here can be applied to any class of graphs.
Section snippets
Methods and results
The concept of the orbit polynomial was introduced in [1]. In this section we calculate this polynomial for graphs with a given number of vertex orbits [4], [19]. Clearly, the number of vertices in each orbit, as well as the multiplicity of orbits of a given cardinality, can vary. In [1], the polynomial derived from the orbit polynomial OG(z), has been used to measure the symmetry of graphs. The positive zero δ of the polynomial was found to be a useful symmetry measure, see [1]
Summary and conclusion
This paper has addressed the problem of determining bounds on the unique positive roots of certain graph polynomials. These polynomials capture the orbit structure of the automorphism group of a graph, and as such give information about symmetry. The results suggest that the roots of these polynomial can be used to classify graphs according to degree of symmetry. Specific bounds have been given for several classes of graphs. Also, we applied the measures δ to special sets of isomers. The
Acknowledgments
Matthias Dehmer thanks the Austrian Science Fund for supporting this work (project P30031).
Zengqiang Chen was supported by National Natural Science Foundation of China (No. 61573199). Jin Tao was supported by Academy of Finland (No. 315660).
Modjtaba Ghorbani was supported by the Shahid Rajaee Teacher Training University under grant No. 22126.
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