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In this paper, we prove bounds for the unique, positive zero of O  G (z) := 1 −O G (z) , where O G ( z ) is the so-called orbit polynomial [1]. The orbit polynomial is based on the multiplic- ity and cardinalities of the vertex orbits of a graph. In [1] , we have shown that the unique, positive zero δ≤1 of O  G (z) can serve as a meaningful measure of graph symmetry. In this paper, we study special graph classes with a specified number of orbits and obtain bounds on the value of δ.

Research on the structural complexity of networks has produced many useful results in graph theory and applied disciplines such as engineering and data analysis. This paper is intended as a further contribution to this area of research. Here we focus on measures designed to compare graphs with respect to symmetry. We do this by means of a novel characteristic of a graph G, namely an ``orbit polynomial.'' A typical term of this univariate polynomial is of the form czn, where c is the number of orbits of size n of the automorphism group of G. Subtracting the orbit polynomial from 1 results in another polynomial that has a unique positive root, which can serve as a relative measure of the symmetry of a graph. The magnitude of this root is indicative of symmetry and can thus be used to compare graphs with respect to that property. In what follows, we will prove several inequalities on the unique positive roots of orbit polynomials corresponding to different graphs, thus showing differences in symmetry. In addition, we present numerical results relating to several classes of graphs for the purpose of comparing the new symmetry measure with existing ones. Finally, it is applied tomore »a set of isomers of the chemical compound adamantane C10H16. We believe that the measure can be quite useful for tackling applications in chemistry, bioinformatics, and structure-oriented drug design.« less

Online consumer reviews contain rich yet implicit information regarding consumers’ preferences for specific aspects of products/services. Extracting aspects from online consumer reviews has been recognized as a valuable step in performing fine-grained analytical tasks (e.g. aspect-based sentiment analysis). Extant approaches to aspect extraction are dominated by discrete models. Despite explosive research interests in continuous-space language models in recent years, these models have yet to be explored for the task of extracting product/service aspects from online consumer reviews. In addition, previous continuous-space models for information extraction have largely overlooked the role of semantic information embedded in texts. In this study, we propose an approach of aspect extraction that leverages semantic information from WordNet in conjunction of building continuous-space language models from review texts. The experiment results with online restaurant reviews demonstrate that the WordNet-guided continuous-space language models outperform both discrete models and continuous-space language models without incorporating the semantic information. The research findings have important implications for understanding consumer preferences and improving business performances.