Distinguishing labeling of group actions

AbstractSuppose Γ is a group acting on a set X. An r-labeling f:X→{1,2,…,r} of X is distinguishing (with respect to Γ) if the only label preserving permutation of X in Γ is the identity. The distinguishing number, DΓ(X), of the action of Γ on X is the minimum r for which there is an r-labeling which is distinguishing. This paper investigates the relation between the cardinality of a set X and the distinguishing numbers of group actions on X. For a positive integer n, let D(n) be the set of distinguishing numbers of transitive group actions on a set X of cardinality n, i.e., D(n)={DΓ(X):|X|=n and Γ acts transitively on X}. We prove that |D(n)|=O(n). Then we consider the problem of an arbitrary fixed group Γ acting on a large set. We prove that if for any action of Γ on a set Y, for each proper normal subgroup H of Γ, DH(Y)≤2, then there is an integer n such that for any set X with |X|≥n, for any action of Γ on X with no fixed points, DΓ(X)≤2

Distinguishing Number for some Circulant Graphs

Introduced by Albertson et al. \cite{albertson}, the distinguishing number $D(G)$ of a graph $G$ is the least integer $r$ such that there is a $r$-labeling of the vertices of $G$ that is not preserved by any nontrivial automorphism of $G$. Most of graphs studied in literature have 2 as a distinguishing number value except complete, multipartite graphs or cartesian product of complete graphs depending on $n$. In this paper, we study circulant graphs of order $n$ where the adjacency is defined using a symmetric subset $A$ of $\mathbb{Z}_n$, called generator. We give a construction of a family of circulant graphs of order $n$ and we show that this class has distinct distinguishing numbers and these lasters are not depending on $n$

Distinguishing Topical and Social Groups Based on Common Identity and Bond Theory

Social groups play a crucial role in social media platforms because they form the basis for user participation and engagement. Groups are created explicitly by members of the community, but also form organically as members interact. Due to their importance, they have been studied widely (e.g., community detection, evolution, activity, etc.). One of the key questions for understanding how such groups evolve is whether there are different types of groups and how they differ. In Sociology, theories have been proposed to help explain how such groups form. In particular, the common identity and common bond theory states that people join groups based on identity (i.e., interest in the topics discussed) or bond attachment (i.e., social relationships). The theory has been applied qualitatively to small groups to classify them as either topical or social. We use the identity and bond theory to define a set of features to classify groups into those two categories. Using a dataset from Flickr, we extract user-defined groups and automatically-detected groups, obtained from a community detection algorithm. We discuss the process of manual labeling of groups into social or topical and present results of predicting the group label based on the defined features. We directly validate the predictions of the theory showing that the metrics are able to forecast the group type with high accuracy. In addition, we present a comparison between declared and detected groups along topicality and sociality dimensions.Comment: 10 pages, 6 figures, 2 table