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A Survey on Symmetry Group of Polyhedral Graphs

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A Survey on Symmetry Group of Polyhedral Graphs S S symmetry Article A Survey on Symmetry Group of Polyhedral Graphs Modjtaba Ghorbani 1,∗ , Matthias Dehmer 2,3,4, Shaghayegh Rahmani 1 and Mina Rajabi-Parsa1 1 Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, 15614 Tehran, Iran; [email protected] (S.R.); [email protected] (M.R.-P.) 2 Steyr School of Management, University of Applied Sciences Upper Austria, 4040 Steyr, Austria; [email protected] 3 Department of Mechatronics and Biomedical Computer Science, UMIT, A-6060 Hall in Tyrol, Austria 4 College of Artficial Intelligence, Nankai University, Tianjin 300071, China * Corresponding: [email protected]; Tel.: +98-21-22970029 Received: 13 February 2020; Accepted: 21 February 2020; Published: 2 March 2020 Abstract: Every three-connected simple planar graph is a polyhedral graph and a cubic polyhedral graph with pentagonal and hexagonal faces is called as a classical fullerene. The aim of this paper is to survey some results about the symmetry group of cubic polyhedral graphs. We show that the order of symmetry group of such graphs divides 240. Keywords: polyhedral graph; fullerene; automorphism group; group action 1. Introduction In the present work, all graphs are connected without loops and parallel edges, which we call simple graphs. An automorphism of a graph is a permutation on the set of vertices which preserves the edge set. We denote the image of automorphism b of graph G on the vertex u by b(u). The set Aut(G) = fa(u) : a is an automorphism g with the operation of composition is a permutation group on V(G) and we call it as automorphism group of G. The first mathematician who considered the graph automorphism was Frucht. In addition, the numerical measures based on automorphisms of a graph have been investigated in reference [1]. A fullerene is a cubic three-connected graph whose faces are pentagons and hexagons. All three connected cubic planar graphs with hexagons and pentagons are called as fullerenes and we donote them by PH-fullerenes, see [2,3]. On the other hand, a three-connected cubic planar graph whose faces are triangles and hexagons is denoted by a TH-fullerene and a SH-fullerene is a three connected cubic planar graph with quadrangles and hexagons. See the references [4–6] as well as [7–11], for studying problems concerning with fullerene graphs. In mathematical aspects, fullerenes are member of a big family of larger graphs, namely the polyhedral graphs. In general, a polyhedral graph is three connected planar but in this paper, we just consider cubic polyhedral graphs, see [12–15]. We denote a cubic polyhedral graph with t triangles, s quadrangles, p pentagonal and h hexagonal faces and no other faces by a (t, s, p, h)−polyhedral or briefly a (t, s, p)-polyhedral graph. This yields that in a SPH-polyhedral graph all faces are tetragons, pentagons and hexagons. Here, we enumerate the number of edges of a SPH-polyhedral graph F. It is clear that each vertex lies in three faces, because F is 3-regular. In addition, we enumerate each edge two times. Hence, jVj = (4s + 5p + 6h)/3, jEj = (4s + 5p + 6h)/2 = 3n/2 and the number of faces is f = s + p + h. The Euler’s formula yields that n − m + f = 2 and so (4s + 5p + 6h)/3 − (4s + 5p + 6h)/2 + s + h + p = 2. This means that p = 12 − 2s and h = n/2 + s − 10. Since p ≥ 0 we get s ≤ 6 and the following cases hold: Case 1. If s = 0, then F is a fullerene and jAut(F)j divides 120. Symmetry 2020, 12, 370; doi:10.3390/sym12030370 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 370 2 of 30 Cases 2–6. If s is either the number 1 or 2 or 3 or 4 or 5, then p is 10 or 8 or 6 or 4 or 2, respectively. Case 7. If s = 6, then one can deduce that p = 0 and thus F is a SPH-fullerene and jAut(F)j divides 24. By above notation, we denote a polyhedral graph with triangles, pentagons and hexagons by a TPH-polyhedral graph. Suppose t is the number of triangles in F, then above discussion yields that p = 12 − 3s, h = n/2 + s − 10 and s ≤ 4. If s = 0, then F is a fullerene. If s is one the integers 1 or 2 or 3, then p is 9 or 6 or 3, respectively. If s = 4, then p = 0 and thus F is a TH-fullerene. One can easily see that in a TSH-polyhedral graph with t triangles, s squares and h hexagons, we yield that n = (3t + 4s + 6h)/3, m = (3t + 4s + 6h)/2 and f = t + s + h. Consequently, we obtain either s = 0 or s = 3 or s = 6. If s = 0, then F is TH-fullerene. If s = 6, then t = 0 and F is SH-fullerene. If s = 3, then t = 2 and F is a polyhedral graph with exactly two triangles, three squares and h hexagons. The cube graph Q3 is the smallest SPH-polyhedral graph which has no hexagonal face, see Figure1a. The symmetry group of this graph is isomorphic to Z2 × S4, see [16]. The pyramid is the smallest TPH-polyhedral graph, see Figure1b. The smallest TSH-polyhedral graph has no hexagon, see Figure1c. Figure 1. The smallest SPH, TPH and TSH polyhedral graphs. Much research about fullerene was started after producing fullerenes in bulk quantities in 1990, see [17]. Fullerene chemistry is nowadays a well-established field of both theoretical and experimental investigations. The initial enchanting appeal of fullerenes goes back to the high symmetry of these carbon nanostructures, see [18,19] but nowadays the fullerene era is a field of both theoretical and experimental research. The most important problem, with many applications in a large number of area, is finding the symmetry of molecules. There are several ways to determine the symmetry of a molecule. For example, Randi´c[20,21] and then Balasubramanian [22–29] studied the Euclidean matrix of a chemical graph to find the symmetry group. However, there is a classic theorem in algebraic graph theory which yields the automorphism group of each graph. Suppose s 2 Sn is an arbitrary permutation. Then, the permutation matrix Ps is defined as Ps = [xij], where xij = 1 if i = s(j) and 0, otherwise. It is easy to see that the set of all n × n permutation matrices is a group isomorphic to the symmetric group Sn on n symbols. Moreover, a permutation s 2 Sn is a graph automorphism, if it satisfies PsA = APs, where A is the adjacency matrix of G. In general, the symmetry group and the point-group symmetry of a graph are not isomorphic but about the regular polehedral graphs they are the same. By usung this fact, the authors of [30] computed the symmetry of all fullerenes with up to 70 vertices. In the present paper, we deal with the symmetry properties of fullerene graphs and we compute the symmetry group of several infinite classes of fullerene graphs. One of the aim of computing the symmetry of fullerenes is to enumerate the number of isomers whose number increases very quickly with n. A number of papers deal with the symmetry of fullerene isomers and related species [31–33]. The first author computed the automorphism group of some infinite families of fullerene graphs by using GAP programs [34]. Here, we improve the mentioned algorithm to compute the automorphism group of these fullerene graphs. Further, this paper bears some novel results significant for algorithmization of the fullerene graph machine analysis. Symmetry 2020, 12, 370 3 of 30 2. Main Results A polyhedral graph with either isolated squares or isolated pentagons is called respectively as ISR or IPR polyhedral graph. Here, we consider only the ISR and IPR polyhedral graphs. In [7] Fowler et al. showed that there are only 28 point groups that a fullerene can be realized. In [35] the author investigated that for a PH-fullerene graph F, jAut(F)j divides 120. Similar results obtained by Ghorbani et al. in [36,37] who showed that if F is a TH-polyhedral graph or a SH-polyhedral graph, then jAut(F)j divides 24 or 48, respectively. They also determined the order of automorphism groups of all TSH, TPH and SPH-polyhedral graphs, see [38]. In other words, they proved the following results. Proposition 1. [38] If F is both ISR and IPR, and it is either an SPH or TSH or TPH-polyhedral graph, then for the vertex u 2 V(F), we yield that the stabilizer Aut(F)u is trivial or it is isomorphic to one of three groups: the cyclic group Z2, the cyclic group Z3 and the symmetric group S3. Theorem 1. [38] Let F be both ISR and IPR, SPH-polyhedral graph, G be a TSH-polyhedral graph and L be a TPH-polyhedral graph. Then Aut(F) is a f2, 3, 5g-group and both Aut(G) and Aut(L) are f2, 3g-group. Moreover, jAut(F)j divides 24 × 3 × 5, jAut(G)j divides 24 × 3 and jAut(L)j divides 23 × 3. In [13] Deza et al. investigated the list of allowed symmetry groups of smallest polyhedral graphs, see [7,14,15]. Theorem 2. Fora cubic (t, s, p)-polyhedral graphs, the possible point groups and the number of vertices of the smallest one are i. (t, s, p) = (4, 0, 0): Z2 × Z2, Z2 × Z2 × Z2, D8, A4, S4.
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