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Journal articles on the topic 'Automorphisme des graphes'

Author: Grafiati

Published: 20 April 2024

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1

Kutnar, Klavdija, Dragan Marusic, Stefko Miklavic, and Rok Strasek. "Automorphisms of Tabacjn graphs." Filomat 27, no. 7 (2013): 1157–64. http://dx.doi.org/10.2298/fil1307157k.

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A bicirculant is a graph admitting an automorphism whose cyclic decomposition consists of two cycles of equal length. In this paper we consider automorphisms of the so-called Tahacjn graphs, a family of pentavalent bicirculants which are obtained from the generalized Petersen graphs by adding two additional perfect matchings between the two orbits of the above mentioned automorphism. As a corollary, we determine which Tabacjn graphs are vertex-transitive.

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2

Della-Giustina, James. "Finding the Fixing Number of Johnson Graphs J(n, k) for k Є {2; 3}." American Journal of Undergraduate Research 20, no. 3 (December 31, 2023): 81–89. http://dx.doi.org/10.33697/ajur.2023.097.

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The graph invariant, aptly named the fixing number, is the smallest number of vertices that, when fixed, eliminate all non-trivial automorphisms (or symmetries) of a graph. Although many graphs have established fixing numbers, Johnson graphs, a family of graphs related to the graph isomorphism problem, have only partially classified fixing numbers. By examining specific orbit sizes of the automorphism group of Johnson graphs and classifying the subsequent remaining subgroups of the automorphism group after iteratively fixing vertices, we provide exact minimal sequences of fixed vertices, in turn establishing the fixing number of infinitely many Johnson graphs. KEYWORDS: Graph Automorphism Groups; Symmetry Breaking; Fixing Number; Determining Number; Johnson Graphs; Kneser Graphs; Graph Invariants; Permutation Groups; Minimal Sized Bases.

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3

Ghorbani, Modjtaba, Matthias Dehmer, Abbe Mowshowitz, Jin Tao, and Frank Emmert-Streib. "The Hosoya Entropy of Graphs Revisited." Symmetry 11, no. 8 (August 6, 2019): 1013. http://dx.doi.org/10.3390/sym11081013.

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In this paper we extend earlier results on Hosoya entropy (H-entropy) of graphs, and establish connections between H-entropy and automorphisms of graphs. In particular, we determine the H-entropy of graphs whose automorphism group has exactly two orbits, and characterize some classes of graphs with zero H-entropy.

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Maksimović, Marija. "On Some Regular Two-Graphs up to 50 Vertices." Symmetry 15, no. 2 (February 3, 2023): 408. http://dx.doi.org/10.3390/sym15020408.

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Regular two-graphs on up to 36 vertices are classified, and recently, the classification of regular two-graphs on 38 and 42 vertices having at least one descendant with a nontrivial automorphism group has been performed. The first unclassified cases are those on 46 and 50 vertices. It is known that there are at least 97 regular two-graphs on 46 vertices leading to 2104 descendants and 54 regular two-graphs on 50 vertices leading to 785 descendants. In this paper, we classified all strongly regular graphs with parameters (45,22,10,11), (49,24,11,12), and (50,21,8,9) that have Z6 as the automorphism group and constructed regular two-graphs from SRGs (45,22,10,11), SRGs (49,24,11,12), and SRGs (50,21,8,9) that have automorphisms of order six. In this way, we enumerated all regular two-graphs on up to 50 vertices that have at least one descendant with an automorphism group of order six or at least one strongly regular graph associated with an automorphism group of order six. We found 236 new regular two-graphs on 46 vertices leading to 3172 new SRG (45,22,10,11) and 51 new regular two-graphs on 50 vertices leading to 398 new SRG (49,24,11,12).

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5

Łuczak, Tomasz. "The automorphism group of random graphs with a given number of edges." Mathematical Proceedings of the Cambridge Philosophical Society 104, no. 3 (November 1988): 441–49. http://dx.doi.org/10.1017/s0305004100065646.

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An automorphism σ(G) of a graph G is a permutation of the set of its vertices which preserves adjacency. Under the operation of composition the automorphisms of G form a group Aut(G). The graph G is called asymmetric if Aut(G) is trivial, and symmetric otherwise.

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6

Hernández-Gómez, Juan C., Gerardo Reyna-Hérnandez, Jesús Romero-Valencia, and Omar Rosario Cayetano. "Transitivity on Minimum Dominating Sets of Paths and Cycles." Symmetry 12, no. 12 (December 11, 2020): 2053. http://dx.doi.org/10.3390/sym12122053.

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Transitivity on graphs is a concept widely investigated. This suggest to analyze the action of automorphisms on other sets. In this paper, we study the action on the family of γ-sets (minimum dominating sets), the graph is called γ-transitive if given two γ-sets there exists an automorphism which maps one onto the other. We deal with two families: paths Pn and cycles Cn. Their γ-sets are fully characterized and the action of the automorphism group on the family of γ-sets is fully analyzed.

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7

Ball, Fabian, and Andreas Geyer-Schulz. "Invariant Graph Partition Comparison Measures." Symmetry 10, no. 10 (October 15, 2018): 504. http://dx.doi.org/10.3390/sym10100504.

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Symmetric graphs have non-trivial automorphism groups. This article starts with the proof that all partition comparison measures we have found in the literature fail on symmetric graphs, because they are not invariant with regard to the graph automorphisms. By the construction of a pseudometric space of equivalence classes of permutations and with Hausdorff’s and von Neumann’s methods of constructing invariant measures on the space of equivalence classes, we design three different families of invariant measures, and we present two types of invariance proofs. Last, but not least, we provide algorithms for computing invariant partition comparison measures as pseudometrics on the partition space. When combining an invariant partition comparison measure with its classical counterpart, the decomposition of the measure into a structural difference and a difference contributed by the group automorphism is derived.

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8

FERN, LORI, GARY GORDON, JASON LEASURE, and SHARON PRONCHIK. "Matroid Automorphisms and Symmetry Groups." Combinatorics, Probability and Computing 9, no. 2 (March 2000): 105–23. http://dx.doi.org/10.1017/s0963548399004125.

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For a subgroup W of the hyperoctahedral group On which is generated by reflections, we consider the linear dependence matroid MW on the column vectors corresponding to the reflections in W. We determine all possible automorphism groups of MW and determine when W ≅ = Aut(MW). This allows us to connect combinatorial and geometric symmetry. Applications to zonotopes are also considered. Signed graphs are used as a tool for constructing the automorphisms.

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9

Moreira de Oliveira, Montauban, and Jean-Guillaume Eon. "Non-crystallographic nets: characterization and first steps towards a classification." Acta Crystallographica Section A Foundations and Advances 70, no. 3 (March 12, 2014): 217–28. http://dx.doi.org/10.1107/s2053273314000631.

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Non-crystallographic (NC) nets are periodic nets characterized by the existence of non-trivial bounded automorphisms. Such automorphisms cannot be associated with any crystallographic symmetry in realizations of the net by crystal structures. It is shown that bounded automorphisms of finite order form a normal subgroupF(N) of the automorphism group of NC nets (N,T). As a consequence, NC nets are unstable nets (they display vertex collisions in any barycentric representation) and, conversely, stable nets are crystallographic nets. The labelled quotient graphs of NC nets are characterized by the existence of an equivoltage partition (a partition of the vertex set that preserves label vectors over edges between cells). A classification of NC nets is proposed on the basis of (i) their relationship to the crystallographic net with a homeomorphic barycentric representation and (ii) the structure of the subgroupF(N).

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10

Tsiovkina, Ludmila Yu. "ON A CLASS OF EDGE-TRANSITIVE DISTANCE-REGULAR ANTIPODAL COVERS OF COMPLETE GRAPHS." Ural Mathematical Journal 7, no. 2 (December 30, 2021): 136. http://dx.doi.org/10.15826/umj.2021.2.010.

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The paper is devoted to the problem of classification of edge-transitive distance-regular antipodal covers of complete graphs. This extends the classification of those covers that are arc-transitive, which has been settled except for some tricky cases that remain to be considered, including the case of covers satisfying condition \(c_2=1\) (which means that every two vertices at distance 2 have exactly one common neighbour).Here it is shown that an edge-transitive distance-regular antipodal cover of a complete graph with \(c_2=1\) is either the second neighbourhood of a vertex in a Moore graph of valency 3 or 7, or a Mathon graph, or a half-transitive graph whose automorphism group induces an affine \(2\)-homogeneous group on the set of its fibres. Moreover, distance-regular antipodal covers of complete graphs with \(c_2=1\) that admit an automorphism group acting \(2\)-homogeneously on the set of fibres (which turns out to be an approximation of the property of edge-transitivity of such cover), are described. A well-known correspondence between distance-regular antipodal covers of complete graphs with \(c_2=1\) and geodetic graphs of diameter two that can be viewed as underlying graphs of certain Moore geometries, allows us to effectively restrict admissible automorphism groups of covers under consideration by combining Kantor's classification of involutory automorphisms of these geometries together with the classification of finite 2-homogeneous permutation groups.

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11

Brooks, Josephine, Alvaro Carbonero, Joseph Vargas, Rigoberto Flórez, Brendan Rooney, and Darren Narayan. "Removing Symmetry in Circulant Graphs and Point-Block Incidence Graphs." Mathematics 9, no. 2 (January 14, 2021): 166. http://dx.doi.org/10.3390/math9020166.

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An automorphism of a graph is a mapping of the vertices onto themselves such that connections between respective edges are preserved. A vertex v in a graph G is fixed if it is mapped to itself under every automorphism of G. The fixing number of a graph G is the minimum number of vertices, when fixed, fixes all of the vertices in G. The determination of fixing numbers is important as it can be useful in determining the group of automorphisms of a graph-a famous and difficult problem. Fixing numbers were introduced and initially studied by Gibbons and Laison, Erwin and Harary and Boutin. In this paper, we investigate fixing numbers for graphs with an underlying cyclic structure, which provides an inherent presence of symmetry. We first determine fixing numbers for circulant graphs, showing in many cases the fixing number is 2. However, we also show that circulant graphs with twins, which are pairs of vertices with the same neighbourhoods, have considerably higher fixing numbers. This is the first paper that investigates fixing numbers of point-block incidence graphs, which lie at the intersection of graph theory and combinatorial design theory. We also present a surprising result-identifying infinite families of graphs in which fixing any vertex fixes every vertex, thus removing all symmetries from the graph.

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12

Ye, Kaidi. "Quotient and blow-up of automorphisms of graphs of groups." International Journal of Algebra and Computation 28, no. 05 (August 2018): 733–58. http://dx.doi.org/10.1142/s0218196718500340.

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In this paper, we study the quotient and “blow-up” of graph-of-groups [Formula: see text] and of their automorphisms [Formula: see text]. We show that the existence of such a blow-up of any [Formula: see text], relative to a given family of “local” graph-of-groups isomorphisms [Formula: see text] depends crucially on the [Formula: see text]-conjugacy class of the correction term [Formula: see text] for any edge [Formula: see text] of [Formula: see text], where [Formula: see text]-conjugacy is a new but natural concept introduced here. As an application, we obtain a criterion as to whether a partial Dehn twist can be blown up relative to local Dehn twists, to give an actual Dehn twist. The results of this paper are also used crucially in the follow-up papers [Lustig and Ye, Normal form and parabolic dynamics for quadratically growing automorphisms of free groups, arXiv:1705.04110v2; Ye, Partial Dehn twists of free groups relative to local Dehn twists — A dichotomy, arXiv:1605.04479 ; When is a polynomially growing automorphism of [Formula: see text] geometric, arXiv:1605.07390 ].

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13

Hernández, Juan C., José M. Rodríguez, and José M. Sigarreta. "Mathematical Properties of the Hyperbolicity of Circulant Networks." Advances in Mathematical Physics 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/723451.

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IfXis a geodesic metric space andx1,x2,x3∈X, ageodesic triangle T={x1,x2,x3}is the union of the three geodesics[x1x2],[x2x3], and[x3x1]inX. The spaceXisδ-hyperbolic(in the Gromov sense) if any side ofTis contained in aδ-neighborhood of the union of the two other sides, for every geodesic triangleTinX. The study of the hyperbolicity constant in networks is usually a very difficult task; therefore, it is interesting to find bounds for particular classes of graphs. A network is circulant if it has a cyclic group of automorphisms that includes an automorphism taking any vertex to any other vertex. In this paper we obtain several sharp inequalities for the hyperbolicity constant of circulant networks; in some cases we characterize the graphs for which the equality is attained.

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14

Csontóová, Mária. "Graph automorphisms of multilattices." Mathematica Bohemica 128, no. 2 (2003): 209–13. http://dx.doi.org/10.21136/mb.2003.134035.

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15

CLAY, MATT, and ALEXANDRA PETTET. "RELATIVE TWISTING IN OUTER SPACE." Journal of Topology and Analysis 04, no. 02 (June 2012): 173–201. http://dx.doi.org/10.1142/s1793525312500100.

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Subsurface projection is indispensable to studying the geometry of the mapping class group and the curve complex of a surface. When the subsurface is an annulus, this projection is sometimes called relative twisting. We give two alternate versions of relative twisting for the outer automorphism group of a free group. We use this to describe sufficient conditions for when a folding path enters the thin part of Culler–Vogtmann's Outer space. As an application of our condition, we produce a sequence of fully irreducible outer automorphisms whose axes in Outer space travel through graphs with arbitrarily short cycles; we also describe the asymptotic behavior of their translation lengths.

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16

Du, Jiali, Yanquan Feng, and Yuqin Liu. "Heptavalent Symmetric Graphs with Certain Conditions." Algebra Colloquium 28, no. 02 (May 11, 2021): 243–52. http://dx.doi.org/10.1142/s1005386721000195.

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A graph [Formula: see text] is said to be symmetric if its automorphism group [Formula: see text] acts transitively on the arc set of [Formula: see text]. We show that if [Formula: see text] is a finite connected heptavalent symmetric graph with solvable stabilizer admitting a vertex-transitive non-abelian simple group [Formula: see text] of automorphisms, then either [Formula: see text] is normal in [Formula: see text], or [Formula: see text] contains a non-abelian simple normal subgroup [Formula: see text] such that [Formula: see text] and [Formula: see text] is explicitly given as one of 11 possible exceptional pairs of non-abelian simple groups. If [Formula: see text] is arc-transitive, then [Formula: see text] is always normal in [Formula: see text], and if [Formula: see text] is regular on the vertices of [Formula: see text], then the number of possible exceptional pairs [Formula: see text] is reduced to 5.

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17

Guo, Songtao, Yantao Li, and Xiaohui Hua. "(G,s)-Transitive Graphs of Valency 7." Algebra Colloquium 23, no. 03 (June 20, 2016): 493–500. http://dx.doi.org/10.1142/s100538671600047x.

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Let X be a finite simple undirected graph and G an automorphism group of X. If G is transitive on s-arcs but not on (s+1)-arcs then X is called (G,s)-transitive. Let X be a connected (G,s)-transitive graph of a prime valency p, and Gv the vertex stabilizer of a vertex v ∈ V(X) in G. For the case p=3, the exact structure of Gv has been determined by Djoković and Miller in [Regular groups of automorphisms of cubic graphs, J. Combin. Theory (Ser. B) 29 (1980) 195 – 230]. For the case p=5, all the possibilities of Gv have been given by Guo and Feng in [A note on pentavalent s-transitive graphs, Discrete Math.312 (2012) 2214 – 2216]. In this paper, we deal with the case p=7 and determine the exact structure of the vertex stabilizer Gv.

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18

Van Bon, John. "Affine distance-transitive graphs with quadratic forms." Mathematical Proceedings of the Cambridge Philosophical Society 112, no. 3 (November 1992): 507–17. http://dx.doi.org/10.1017/s0305004100071188.

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The classification of all finite primitive distance-transitive graphs is basically divided into two cases. In the one case, known as the almost simple case, we have an almost simple group acting primitively as a group of automorphisms on the graph. In the other case, known as the affine case, the vertices of the graph can be identified with the vectors of a finite-dimensional vector space over some finite field. In this case the automorphism group G of the graph Γ contains a normal p-subgroup N which is elementary Abelian and acts regularly on the set of vertices of Γ. Let G0 be the subgroup of G that stabilizes a vertex. Identifying the vertices of Γ with G0-cosets in G, one obtains a vector space V on which N acts as a group of translations, G0, stabilizes 0 and, as Γ is primitive, G0 acts irreducibly on V.

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19

Liu, Ye. "On Chromatic Functors and Stable Partitions of Graphs." Canadian Mathematical Bulletin 60, no. 1 (March 1, 2017): 154–64. http://dx.doi.org/10.4153/cmb-2016-047-3.

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AbstractThe chromatic functor of a simple graph is a functorization of the chromatic polynomial. M. Yoshinaga showed that two ûnite graphs have isomorphic chromatic functors if and only if they have the same chromatic polynomial. The key ingredient in the proof is the use of stable partitions of graphs. The latter is shown to be closely related to chromatic functors. In this note, we further investigate some interesting properties of chromatic functors associated with simple graphs using stable partitions. Our ûrst result is the determination of the group of natural automorphisms of the chromatic functor, which is, in general, a larger group than the automorphism group of the graph. The second result is that the composition of the chromatic functor associated with a ûnite graph restricted to the category FI of ûnite sets and injections with the free functor into the category of complex vector spaces yields a consistent sequence of representations of symmetric groups that is representation stable in the sense of Church–Farb.

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20

Rodríguez, José M., and José M. Sigarreta. "The hyperbolicity constant of infinite circulant graphs." Open Mathematics 15, no. 1 (June 9, 2017): 800–814. http://dx.doi.org/10.1515/math-2017-0061.

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Abstract If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. Deciding whether or not a graph is hyperbolic is usually very difficult; therefore, it is interesting to find classes of graphs which are hyperbolic. A graph is circulant if it has a cyclic group of automorphisms that includes an automorphism taking any vertex to any other vertex. In this paper we prove that infinite circulant graphs and their complements are hyperbolic. Furthermore, we obtain several sharp inequalities for the hyperbolicity constant of a large class of infinite circulant graphs and the precise value of the hyperbolicity constant of many circulant graphs. Besides, we give sharp bounds for the hyperbolicity constant of the complement of every infinite circulant graph.

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21

Jakubík, Ján. "Graph automorphisms of semimodular lattices." Mathematica Bohemica 125, no. 4 (2000): 459–64. http://dx.doi.org/10.21136/mb.2000.126276.

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22

Javaid, Imran, Shahroz Ali, Shahid Ur Rehman, and Aqsa Shah. "Rough sets in graphs using similarity relations." AIMS Mathematics 7, no. 4 (2022): 5790–807. http://dx.doi.org/10.3934/math.2022320.

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<abstract><p>In this paper, we investigate the theory of rough set to study graphs using the concept of orbits. Rough sets are based on a clustering criterion and we use the idea of similarity of vertices under automorphism as a criterion. We introduce indiscernibility relation in terms of orbits and prove necessary and sufficient conditions under which the indiscernibility partitions remain the same when associated with different attribute sets. We show that automorphisms of the graph $ \mathcal{G} $ preserve the indiscernibility partitions. Further, we prove that for any graph $ \mathcal{G} $ with $ k $ orbits, any reduct $ \mathcal{R} $ consists of one element from $ k-1 $ orbits of the graph. We also study the rough membership functions for paths, cycles, complete and complete bipartite graphs. Moreover, we introduce essential sets and discernibility matrices induced by orbits of graphs and study their relationship. We also prove that every essential set consists of union of any two orbits of the graph.</p></abstract>

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23

KhudaBukhsh, Wasiur R., Arnab Auddy, Yann Disser, and Heinz Koeppl. "Approximate lumpability for Markovian agent-based models using local symmetries." Journal of Applied Probability 56, no. 3 (September 2019): 647–71. http://dx.doi.org/10.1017/jpr.2019.44.

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AbstractWe study a Markovian agent-based model (MABM) in this paper. Each agent is endowed with a local state that changes over time as the agent interacts with its neighbours. The neighbourhood structure is given by a graph. Recently, Simon, Taylor, and Kiss [40] used the automorphisms of the underlying graph to generate a lumpable partition of the joint state space, ensuring Markovianness of the lumped process for binary dynamics. However, many large random graphs tend to become asymmetric, rendering the automorphism-based lumping approach ineffective as a tool of model reduction. In order to mitigate this problem, we propose a lumping method based on a notion of local symmetry, which compares only local neighbourhoods of vertices. Since local symmetry only ensures approximate lumpability, we quantify the approximation error by means of the Kullback–Leibler divergence rate between the original Markov chain and a lifted Markov chain. We prove the approximation error decreases monotonically. The connections to fibrations of graphs are also discussed.

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24

Ibarra, Sofía, and Luis Manuel Rivera. "The automorphism groups of some token graphs." Proyecciones (Antofagasta) 42, no. 6 (December 1, 2023): 1627–51. http://dx.doi.org/10.22199/issn.0717-6279-5954.

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The token graphs of graphs have been studied at least from the 80’s with different names and by different authors. The Johnson graph J(n, k) is isomorphic to the k-token graph of the complete graph Kn. To our knowledge, the unique results about the automorphism groups of token graphs are for the case of the Johnson graphs. In this paper we begin the study of the automorphism groups of token graphs of another graphs. In particular we obtain the automorphism group of the k-token graph of the path graph Pn, for n 6≠ 2k. Also, we obtain the automorphism group of the 2-token graph of the following graphs: cycle, star, fan and wheel graphs.

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25

DUNCAN, ANDREW J., and VLADIMIR N. REMESLENNIKOV. "AUTOMORPHISMS OF PARTIALLY COMMUTATIVE GROUPS II: COMBINATORIAL SUBGROUPS." International Journal of Algebra and Computation 22, no. 07 (November 2012): 1250074. http://dx.doi.org/10.1142/s0218196712500749.

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We define several "standard" subgroups of the automorphism group Aut (G) of a partially commutative (right-angled Artin) group and use these standard subgroups to describe decompositions of Aut (G). If C is the commutation graph of G, we show how Aut (G) decomposes in terms of the connected components of C: obtaining a particularly clear decomposition theorem in the special case where C has no isolated vertices. If C has no vertices of a type we call dominated then we give a semi-direct decomposition of Aut (G) into a subgroup of locally conjugating automorphisms by the subgroup stabilizing a certain lattice of "admissible subsets" of the vertices of C. We then characterize those graphs for which Aut (G) is a product (not necessarily semi-direct) of two such subgroups.

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Agafonova, T. A., and V. F. Gor'kovoi. "Automorphisms of graphs." Journal of Mathematical Sciences 72, no. 5 (December 1994): 3341–43. http://dx.doi.org/10.1007/bf01261692.

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27

HUANG, ZHAOHONG, JIANGMIN PAN, SUYUN DING, and ZHE LIU. "AUTOMORPHISM GROUPS OF SELF-COMPLEMENTARY VERTEX-TRANSITIVE GRAPHS." Bulletin of the Australian Mathematical Society 93, no. 2 (November 11, 2015): 238–47. http://dx.doi.org/10.1017/s0004972715001197.

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Li et al. [‘On finite self-complementary metacirculants’, J. Algebraic Combin.40 (2014), 1135–1144] proved that the automorphism group of a self-complementary metacirculant is either soluble or has $\text{A}_{5}$ as the only insoluble composition factor, and gave a construction of such graphs with insoluble automorphism groups (which are the first examples of self-complementary graphs with this property). In this paper, we will prove that each simple group is a subgroup (so is a section) of the automorphism groups of infinitely many self-complementary vertex-transitive graphs. The proof involves a construction of such graphs. We will also determine all simple sections of the automorphism groups of self-complementary vertex-transitive graphs of $4$-power-free order.

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28

Lu, Jian, and Zhongxiang Wang. "On the maximum Graovac-Pisanski index of bicyclic graphs." AIMS Mathematics 8, no. 10 (2023): 24914–28. http://dx.doi.org/10.3934/math.20231270.

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<abstract><p>For a simple graph $ G = (V(G), E(G)) $, the Graovac-Pisanski index of $ G $ is defined as</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ GP(G) = \frac{|V(G)|}{2|{\rm{Aut}}(G)|}\sum\limits_{u\in V(G)}\sum\limits_{\alpha\in {\rm{Aut}}(G)}d_G(u,\alpha(u)), $\end{document} </tex-math></disp-formula></p> <p>where $ {\rm{Aut}}(G) $ is the automorphism group of $ G $ and $ d_G(u, v) $ is the length of a shortest path between the two vertices $ u $ and $ v $ in $ G $. Obviously, $ GP(G) = 0 $ if $ G $ has no nontrivial automorphisms. Let $ B_{n}^{3, 3} $ be the graph consisting of two disjoint 3-cycles with a path of length $ n-5 $ joining them. In this article, we prove that among all those $ n $-vertex bicyclic graphs in which every edge lies on at most one cycle, $ B_{n}^{3, 3} $ has the maximum Graovac-Pisanski index.</p></abstract>

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29

Daws, Matthew. "Quantum graphs: Different perspectives, homomorphisms and quantum automorphisms." Communications of the American Mathematical Society 4, no. 5 (February 20, 2024): 117–81. http://dx.doi.org/10.1090/cams/30.

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We undertake a study of the notion of a quantum graph over arbitrary finite-dimensional C ∗ C^* -algebras B B equipped with arbitrary faithful states. Quantum graphs are realised principally as either certain operators on L 2 ( B ) L^2(B) , the quantum adjacency matrices, or as certain operator bimodules over B ′ B’ . We present a simple, purely algebraic approach to proving equivalence between these settings, thus recovering existing results in the tracial state setting. For non-tracial states, our approach naturally suggests a generalisation of the operator bimodule definition, which takes account of (some aspect of) the modular automorphism group of the state. Furthermore, we show that each such “non-tracial” quantum graph corresponds to a “tracial” quantum graph which satisfies an extra symmetry condition. We study homomorphisms (or CP-morphisms) of quantum graphs arising from unital completely positive (UCP) maps, and the closely related examples of quantum graphs constructed from UCP maps. We show that these constructions satisfy automatic bimodule properties. We study quantum automorphisms of quantum graphs, give a definition of what it means for a compact quantum group to act on an operator bimodule, and prove an equivalence between this definition, and the usual notion defined using a quantum adjacency matrix. We strive to give a relatively self-contained, elementary, account, in the hope this will be of use to other researchers in the field.

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Chen, H. Y., H. Han, and Z. P. Lu. "Some semisymmetric graphs arising from finite vector spaces." Journal of Algebra and Its Applications 19, no. 11 (November 11, 2019): 2050216. http://dx.doi.org/10.1142/s0219498820502163.

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A graph is worthy if no two vertices have the same neighborhood. In this paper, we characterize the automorphism groups of unworthy edge-transitive bipartite graphs, and present some worthy semisymmetric graphs arising from vector spaces over finite fields. We also determine the automorphism groups of these graphs.

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Adams, Richard, Janae Dixon, Jennifer Elder, Jamie Peabody, Oscar Vega, and Karen Willis. "Combinatorial Analysis of a Subtraction Game on Graphs." International Journal of Combinatorics 2016 (August 29, 2016): 1–8. http://dx.doi.org/10.1155/2016/1476359.

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We define a two-player combinatorial game in which players take alternate turns; each turn consists of deleting a vertex of a graph, together with all the edges containing such vertex. If any vertex became isolated by a player’s move then it would also be deleted. A player wins the game when the other player has no moves available. We study this game under various viewpoints: by finding specific strategies for certain families of graphs, through using properties of a graph’s automorphism group, by writing a program to look at Sprague-Grundy numbers, and by studying the game when played on random graphs. When analyzing Grim played on paths, using the Sprague-Grundy function, we find a connection to a standing open question about Octal games.

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Pan, Jiangmin. "On finite dual Cayley graphs." Open Mathematics 18, no. 1 (June 22, 2020): 595–602. http://dx.doi.org/10.1515/math-2020-0141.

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Abstract A Cayley graph \Gamma on a group G is called a dual Cayley graph on G if the left regular representation of G is a subgroup of the automorphism group of \Gamma (note that the right regular representation of G is always an automorphism group of \Gamma ). In this article, we study finite dual Cayley graphs regarding identification, construction, transitivity and such graphs with automorphism groups as small as possible. A few problems worth further research are also proposed.

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Mude, Lao Hussein, Owino Maurice Oduor, and Ojiema Michael Onyango. "Automorphisms of Zero Divisor Graphs of Power Four Radical Zero Completely Primary Finite Rings." Asian Research Journal of Mathematics 19, no. 8 (June 19, 2023): 108–13. http://dx.doi.org/10.9734/arjom/2023/v19i8693.

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Let R be a commutative unital finite rings and Z(R) be its set of zero divisors. The study of automorphisms of algebraic structures via zero divisor graphs is still an active area of research. Perhaps, because of the fact that automorphisms have got real life application in capturing the symmetries of algebraic structures. In this study, the automorphisms zero divisor graphs of such rings in which the product of any four zero divisor is zero has been determined.

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FIOL, M. A., E. GARRIGA, and J. L. A. YEBRA. "On Twisted Odd Graphs." Combinatorics, Probability and Computing 9, no. 3 (May 2000): 227–40. http://dx.doi.org/10.1017/s0963548300004181.

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The twisted odd graphs are obtained from the well-known odd graphs through an involutive automorphism. As expected, the twisted odd graphs share some of the interesting properties of the odd graphs but, in general, they seem to have a more involved structure. Here we study some of their basic properties, such as their automorphism group, diameter, and spectrum. They turn out to be examples of the so-called boundary graphs, which are graphs satisfying an extremal property that arises from a bound for the diameter of a graph in terms of its distinct eigenvalues.

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35

Baumann, U. "Automorphisms of Cayley graphs." Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 61, no. 1 (December 1991): 73–81. http://dx.doi.org/10.1007/bf02950754.

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36

Benvenuti, Silvia, and Riccardo Piergallini. "Automorphisms of trivalent graphs." European Journal of Combinatorics 34, no. 6 (August 2013): 987–1009. http://dx.doi.org/10.1016/j.ejc.2013.01.009.

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Čibej, Uroš, and Jurij Mihelič. "Graph automorphisms for compression." Open Computer Science 11, no. 1 (December 17, 2020): 51–59. http://dx.doi.org/10.1515/comp-2020-0186.

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AbstractDetecting automorphisms is a natural way to identify redundant information presented in structured data. When such redundancies are detected they can be used for data compression. In this paper we explore two different classes of graphs to capture this intuitive property of automorphisms. Symmetry-compressible graphs are the first class which introduces the basic concepts but use only global symmetries for the compression. In order for this concept to be more practical, we need to use local symmetries. Thus, we extend the basic graph class with Near Symmetry compressible graphs. Furthermore, we develop two algorithms that can be used to compress practical instances and empirically evaluate them on a set of realistic graphs.

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Joardar, Soumalya, and Arnab Mandal. "Quantum symmetry of graph C∗-algebras associated with connected graphs." Infinite Dimensional Analysis, Quantum Probability and Related Topics 21, no. 03 (September 2018): 1850019. http://dx.doi.org/10.1142/s0219025718500194.

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We define a notion of quantum automorphism groups of graph [Formula: see text]-algebras for finite, connected graphs. Under the assumption that the underlying graph does not have any multiple edge or loop, the quantum automorphism group of the underlying directed graph in the sense of Banica [Quantum automorphism groups of homogeneous graphs, J. Funct. Anal. 224 (2005) 243–280] (which is also the symmetry object in the sense of [S. Schmidt and M. Weber, Quantum symmetry of graph [Formula: see text]-algebras, arXiv:1706.08833 ] is shown to be a quantum subgroup of quantum automorphism group in our sense. Quantum symmetries for some concrete graph [Formula: see text]-algebras have been computed.

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Lotfi, Abdullah, Abbe Mowshowitz, and Matthias Dehmer. "A Note on Eigenvalues and Asymmetric Graphs." Axioms 12, no. 6 (May 24, 2023): 510. http://dx.doi.org/10.3390/axioms12060510.

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This note is intended as a contribution to the study of quantitative measures of graph complexity that use entropy measures based on symmetry. Determining orbit sizes of graph automorphism groups is a key part of such studies. Here we focus on an extreme case where every orbit contains just a single vertex. A permutation of the vertices of a graph G is an automorphism if, and only if, the corresponding permutation matrix commutes with the adjacency matrix of G. This fact establishes a direct connection between the adjacency matrix and the automorphism group. In particular, it is known that if the eigenvalues of the adjacency matrix of G are all distinct, every non-trivial automorphism has order 2. In this note, we add a condition to the case of distinct eigenvalues that makes the graph asymmetric, i.e., reduces the automorphism group to the identity alone. In addition, we prove analogous results for the Google and Laplacian matrices. The condition is used to build an O(n3) algorithm for detecting identity graphs, and examples are given to demonstrate that it is sufficient, but not necessary.

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Ou, Shikun, Yanqi Fan, and Qunfang Li. "Automorphism Group and Other Properties of Zero Component Graph over a Vector Space." Journal of Mathematics 2021 (April 10, 2021): 1–8. http://dx.doi.org/10.1155/2021/5595620.

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In this paper, we introduce an undirected simple graph, called the zero component graph on finite-dimensional vector spaces. It is shown that two finite-dimensional vector spaces are isomorphic if and only if their zero component graphs are isomorphic, and any automorphism of a zero component graph can be uniquely decomposed into the product of a permutation automorphism and a regular automorphism. Moreover, we find the dominating number, as well as the independent number, and characterize the minimum independent dominating sets, maximum independent sets, and planarity of the graph. In the case that base fields are finite, we calculate the fixing number and metric dimension of the zero component graphs and determine vector spaces whose zero component graphs are Hamiltonian.

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Panteleev, Dmitry, and Alexander Ushakov. "Conjugacy search problem and the Andrews–Curtis conjecture." Groups Complexity Cryptology 11, no. 1 (May 1, 2019): 43–60. http://dx.doi.org/10.1515/gcc-2019-2005.

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AbstractWe develop new computational methods for studying potential counterexamples to the Andrews–Curtis conjecture, in particular, Akbulut–Kurby examples {\operatorname{AK}(n)}. We devise a number of algorithms in an attempt to disprove the most interesting counterexample {\operatorname{AK}(3)}. That includes an efficient implementation of the folding procedure for pseudo-conjugacy graphs, based on the original modification of a classic disjoint-set data structure. To improve metric properties of the search space (the set of balanced presentations of the trivial group), we introduce a new transformation, called an ACM-move, that generalizes the original Andrews–Curtis transformations and discuss details of a practical implementation. To reduce growth of the search space, we introduce a strong equivalence relation on balanced presentations and study the space modulo automorphisms of the underlying free group. We prove that automorphism moves can be applied to Akbulut–Kurby presentations. The improved technique allows us to enumerate balanced presentations AC-equivalent to {\operatorname{AK}(3)} with relations of lengths up to 20 (previous record was 17).

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42

Baburin, Igor A. "On Cayley graphs of {\bb Z}^4." Acta Crystallographica Section A Foundations and Advances 76, no. 5 (July 16, 2020): 584–88. http://dx.doi.org/10.1107/s2053273320007159.

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The generating sets of {\bb Z}^4 have been enumerated which consist of integral four-dimensional vectors with components −1, 0, 1 and allow Cayley graphs without edge intersections in a straight-edge embedding in a four-dimensional Euclidean space. Owing to computational restrictions the valency of enumerated graphs has been fixed to 10. Up to isomorphism 58 graphs have been found and characterized by coordination sequences, shortest cycles and automorphism groups. To compute automorphism groups, a novel strategy is introduced that is based on determining vertex stabilizers from the automorphism group of a sufficiently large finite ball cut out from an infinite graph. Six exceptional, rather `dense' graphs have been identified which are locally isomorphic to a five-dimensional cubic lattice within a ball of radius 10. They could be built by either interconnecting interpenetrated three- or four-dimensional cubic lattices and therefore necessarily contain Hopf links between quadrangular cycles. As a consequence, a local combinatorial isomorphism does not extend to a local isotopy.

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Chang, Wen, and Bin Zhu. "Cluster automorphism groups and automorphism groups of exchange graphs." Pacific Journal of Mathematics 307, no. 2 (September 4, 2020): 283–302. http://dx.doi.org/10.2140/pjm.2020.307.283.

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Chen, Jing, Xu Yang, and Xiaomin Zhu. "Isomorphisms and Automorphisms of Generalized Semi-Cayley Graphs." Algebra Colloquium 26, no. 02 (May 7, 2019): 321–28. http://dx.doi.org/10.1142/s1005386719000245.

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In this paper we propose the concept of generalized semi-Cayley graphs, which is a combination of semi-Cayley graphs and generalized Cayley graphs. We study the isomorphisms and automorphisms of generalized semi-Cayley graphs and other related properties.

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45

McColm, Gregory. "Generating Geometric Graphs Using Automorphisms." Journal of Graph Algorithms and Applications 16, no. 2 (2012): 507–41. http://dx.doi.org/10.7155/jgaa.00272.

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Wan, Zhe-Xian, and Kai Zhou. "Unitary Graphs and Their Automorphisms." Annals of Combinatorics 14, no. 3 (September 2010): 367–95. http://dx.doi.org/10.1007/s00026-010-0065-2.

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47

Pankov, Mark. "Automorphisms of infinite Johnson graphs." Discrete Mathematics 313, no. 5 (March 2013): 721–25. http://dx.doi.org/10.1016/j.disc.2012.10.018.

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Bhutani, Kiran R. "On automorphisms of fuzzy graphs." Pattern Recognition Letters 9, no. 3 (April 1989): 159–62. http://dx.doi.org/10.1016/0167-8655(89)90049-4.

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Tang, Zhongming, and Zhe-xian Wan. "Symplectic graphs and their automorphisms." European Journal of Combinatorics 27, no. 1 (January 2006): 38–50. http://dx.doi.org/10.1016/j.ejc.2004.08.002.

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Ghorbani, Modjtaba, Mardjan Hakimi-Nezhaad, Matthias Dehmer, and Xueliang Li. "Analysis of the Graovac–Pisanski Index of Some Polyhedral Graphs Based on Their Symmetry Group." Symmetry 12, no. 9 (August 25, 2020): 1411. http://dx.doi.org/10.3390/sym12091411.

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The Graovac–Pisanski (GP) index of a graph is a modified version of the Wiener index based on the distance between each vertex x and its image α(x), where α is an automorphism of graph. The aim of this paper is to compute the automorphism group of some classes of cubic polyhedral graphs and then we determine their Wiener index. In addition, we investigate the GP-index of these classes of graphs.

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