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Orbit Entropy and Symmetry Index Revisited mathematics Article Orbit Entropy and Symmetry Index Revisited Maryam Jalali-Rad 1 , Modjtaba Ghorbani 1,* , Matthias Dehmer 2,3,4 and Frank Emmert-Streib 5,6 1 Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran 16785-136, Iran; [email protected] 2 Department of Computer Science, Swiss Distance University of Applied Sciences, 3900 Brig, Switzerland; [email protected] 3 Department of Biomedical Computer Science and Mechatronics, UMIT, 6060 Hall in Tyrol, Austria 4 College of Artificial Intelligence, Nankai University, Tianjin 300350, China 5 Predictive Society and Data Analytics Lab, Tampere University, Tampere, Korkeakoulunkatu 10, 33720 Tampere, Finland; frank.emmert-streib@tut.fi 6 Institute of Biosciences and Medical Technology, Tampere University, Tampere, Korkeakoulunkatu 10, 33720 Tampere, Finland * Correspondence: [email protected]; Tel.: +98-21-2297-0005 Abstract: The size of the orbits or similar vertices of a network provides important information regarding each individual component of the network. In this paper, we investigate the entropy or information content and the symmetry index for several classes of graphs and compare the values of this measure with that of the symmetry index of certain graphs. Keywords: entropy; symmetry index; automorphism group 1. Introduction Citation: Jallai-Rad, M.; Ghorbani, Graph entropy measures were first introduced in the study of biological and chemical M.; Dehmer, M.; Emmert-Streib, F. systems, with Rashevsky [1] and Mowshowitz [2–5] making the main contributions. In Orbit Entropy and Symmetry Index particular, Mowshowitz [5] interpreted the topological information content of a graph, such Revisited. Mathematics 2021, 9, 1086. as its entropy measure. Since then, various graph entropy measures have been defined to https://doi.org/10.3390/math9101086 investigate the structural properties of graphs [6–8] as well as [9–13]. A small-world graph [14,15] is a special type of graph in which the neighbors of any Academic Editor: Gabriel Eduard given vertex are likely to be neighbors of each other, but the probability that a vertex is Vilcu the neighbor of another one is low and most vertices can be reached from each other by a few steps. Received: 23 March 2021 Adaptive networks are suitable to model the complex treatment represented by vari- Accepted: 29 April 2021 Published: 12 May 2021 ous real-world systems as well as to carry out decentralized information processing tasks such as drifting conditions and learning from online streaming data, see [16]. On the other hand, signal processing on graphs extends concepts and techniques from traditional signal Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in processing to data indexed by generic graphs, see [17]. For example, neural networks and published maps and institutional affil- graph signal processing have emerged as important actors in data-science applications iations. dealing with complex datasets, see [18]. This paper has two objectives. In Section1, we investigate the automorphism group of some classes of graphs and verify their entropies and symmetry indices. In this way, some practical graph automorphism group decompositions are created that constitute the whole structure of graph automorphism groups. Copyright: © 2021 by the authors. In Section2, we state concepts we use tp perform our analysis. We prove that there are Licensee MDPI, Basel, Switzerland. This article is an open access article several classes of graphs whose symmetry index is greater or equal than the orbit-entropy distributed under the terms and measure, while many other classes have a greater orbit entropy. conditions of the Creative Commons 2. Entropy Measure and Symmetry Index of Graphs Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ Let G = (V(G), E(G)) be a connected graph. An automorphism is a permutation −1 4.0/). a on the set of vertices of G with the property that both a and a preserve the vertex Mathematics 2021, 9, 1086. https://doi.org/10.3390/math9101086 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 1086 2 of 13 adjacency. In other words, for two vertices u, v 2 V(G) a permutation a on V(G) is an automorphism, when uv 2 E(G) if and only if a(u)a(v) 2 E(G), where a(u) is the image of vertex u. The set of all automorphisms under the composition of maps forms a group denoted by Aut(G). For example consider the cycle graph C3 in Figure1. The line of symmetry col- ored by red is denoted by permutation (2, 3). Hence, the permutations correspond to blue and green lines are respectively denoted by (1, 2) and (1, 3). A clockwise rota- ◦ ◦ tion equal 120 around the middle point of C3 is denoted by (1, 2, 3) and equal 240 by (1, 3, 2). All of these permutations preserve the vertex adjacency and thus are auto- morphisms. Note that a 360◦ rotation preserves the figure unchanged and we denote this permutation by (). Hence, the automorphism group of C3 has 6 elements which is Aut(C3) = f(), (1, 2), (1, 3), (2, 3), (1, 2, 3), (1, 3, 2)g. Figure 1. The automorphism group Aut(C3) contains six permutations. For any vertex u 2 V(G) an orbit of G containing u is defined as uG = O(u) = fa(u) : a 2 Aut(G)g. We say G is vertex-transitive if it has only one orbit. Equivalently, a graph is vertex-transitive if for two vertices u, v 2 V(G) there is an automorphism s 2 Aut(G) such that s(u) = v. The orbits of the automorphism group of a graph form a partition of the vertices of the graph. This decomposition introduces the symmetry structure of the graph, and the orbit entropy measure obtained from the automorphism group provides an index of the complexity of the graph relative to the symmetry structure. Mowshowitz [2] defined the topological information content, which is a classical graph entropy measure, as k jO j jO j I (G) = − i log( i ), a ∑ j j j j i=1 V V where Oi (1 ≤ i ≤ k) are orbits of G under the action of automorphism group on the set of vertices. The collection of k orbits fO1, ... , Okg defines a finite probability scheme in an obvious way. This measure is addressed to the problem of measuring the relative complexity of graphs. The idea of measuring the information content of a graph was first presented in [1]. Mowshowitz and Dehmer [19] defined the symmetry index S(G) as S(G) = (log n − Ia(G)) + log jAut(G)j 1 k = (∑ jOij log jOij) + log jAut(G)j. n i=1 2.1. Relationship between Symmetry Index and Orbit Entropy Consider a permutation s on the set X = fx1, ... , xng. Then the set of all elements that s moves is called the support of s. Two permutations s and g are disjoint if their supports have no intersection. Consider S to be a set of generators of Aut(G), e 62 S and S = S1 [ ... [ Sm to be the partition of S, where Si cannot be decomposed into smaller support-disjoint subsets. Therefore, we have the following. Mathematics 2021, 9, 1086 3 of 13 ∼ Theorem 1. Ref. [20] If Ai = hSii, then Aut(G) = A1 × A2 × ... × Am. Consider A = Aut(G). For a vertex v in V(G), Av denotes the stabilizer subgroup of G containing all automorphisms that fix the vertex v. Similarly, for a vertex u of G, its orbit is a set containing all a(u), where a is an automorphism of Aut(G). Theorem 2. Ref. [21](Orbit-stabilizer Theorem) Let A be a permutation group acting on a set A W and u be an arbitrary point in the set W. Then jAj = jAujju j. Definition 1. Harary [22] defined the corona product G1 ◦ G2 of two graphs G1 and G2 as a new graph G obtained by taking one copy of G1 (which has p1 vertices) together with p1 copies of G2 and then joining the ith vertex of G1 to all vertices in the ith copy of G2; see Figure2. Figure 2. G1, G2, G1 ◦ G2, and G2 ◦ G1. Suppose A and B are two finite groups in which B acts on the set X. The wreath product of A and B (denoted by A o B) is a group with the underline set A o B = f( f ; h)j f : X ! A is a function, h 2 Bg. The group operation can be defined as ( f1; h1)( f2; h2) = (g; h1h2), where for each element i 2 X, we obtain h1 g(i) = f1(i) f2(i ). Wreath product is one of the most significant combinatorial buildings in the field of permutation group theory. The next theorem shows that the automorphism group of a big graph can be constructed from wreath product of automorphism groups of its subgraphs. Theorem 3. Ref. [22] The automorphism group of the corona product G1 ◦ G2 of two graphs is isomorphic to the wreath product Aut(G1) o Aut(G2) if and only if either G1 or G2 has no isolated vertices. Let G and H be two disjoint graphs, where V(G) = fu1, ... , ung, and let u 2 V(G) and v 2 V(H). The splice of two graphs G and H by vertices u and v, denoted by G • H(u, v), is a new graph constructed by identifying two vertices u and v in the union of G and H [23]. Similarly, let K be a graph constructed by G and n copies of graph H and then splicing vertex ui of G by vertex v of the ith copy of H.
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