Automorphism Group Of Graph Research Articles

Automorphism group of the reduced power (di)graph of a finite group

We describe the full automorphism group of the directed reduced power graph and the undirected reduced power graph of a finite group. We compute the full automorphism groups of these graphs of several classes of finite groups. Also, we establish some relation between the automorphism group of the undirected reduced power graph (resp. the directed reduced power graph) and the automorphism group of the undirected power graph (resp. the directed power graph) of a finite group.

The graph automorphism group of the dissociation microequilibrium of polyprotic acids

The dissociation micro-states (DMSs) of an $N$-protic acid are described using set theory notation, which facilitates the mathematical description of the dissociation micro-equilibrium (DME). In particular, the DME constants are...

On the structure of consistent cycles in cubic symmetric graphs

AbstractA cycle in a graph is consistent if the automorphism group of the graph admits a one‐step rotation of this cycle. A thorough description of consistent cycles of arc‐transitive subgroups in the full automorphism groups of finite cubic symmetric graphs is given.

Classification of K-forms in nilpotent Lie algebras associated to graphs

Given a simple undirected graph, one can construct from it a c-step nilpotent Lie algebra for every and over any field K, in particular also over the real and complex numbers. These Lie algebras form an important class of examples in geometry and algebra, and it is interesting to link their properties to the defining graph. In this paper, we classify the isomorphism classes of K-forms in these real and complex Lie algebras for any subfield from the structure of the graph. As an application, we show that the number of rational forms up to isomorphism is always one or infinite, with the former being true if and only if the group of graph automorphisms is generated by transpositions.

Perfect Colorings of the Infinite Square Grid: Coverings and Twin Colors

A perfect coloring (equivalent concepts are equitable partition and partition design) of a graph $G$ is a function $f$ from the set of vertices onto some finite set (of colors) such that every node of color $i$ has exactly $S(i,j)$ neighbors of color $j$, where $S(i,j)$ are constants, forming the matrix $S$ called quotient. If $S$ is an adjacency matrix of some simple graph $T$ on the set of colors, then $f$ is called a covering of the target graph $T$ by the cover graph $G$. We characterize all coverings by the infinite square grid, proving that every such coloring is either orbit (that is, corresponds to the orbit partition under the action of some group of graph automorphisms) or has twin colors (that is, two colors such that unifying them keeps the coloring perfect). The case of twin colors is separately classified.

Totally 2-closed finite groups with trivial Fitting subgroup

A finite permutation group [Formula: see text] is called [Formula: see text]-closed if [Formula: see text] is the largest subgroup of [Formula: see text] which leaves invariant each of the [Formula: see text]-orbits for the induced action on [Formula: see text]. Introduced by Wielandt in 1969, the concept of [Formula: see text]-closure has developed as one of the most useful approaches for studying relations on a finite set invariant under a group of permutations of the set; in particular for studying automorphism groups of graphs and digraphs. The concept of total [Formula: see text]-closure switches attention from a particular group action, and is a property intrinsic to the group: a finite group [Formula: see text] is said to be totally [Formula: see text]-closed if [Formula: see text] is [Formula: see text]-closed in each of its faithful permutation representations. There are infinitely many finite soluble totally [Formula: see text]-closed groups, and these have been completely characterized, but up to now no insoluble examples were known. It turns out, somewhat surprisingly to us, that there are exactly [Formula: see text] totally [Formula: see text]-closed finite nonabelian simple groups: the Janko groups [Formula: see text], [Formula: see text] and [Formula: see text], together with [Formula: see text], [Formula: see text] and the Monster [Formula: see text]. Moreover, if a finite totally [Formula: see text]-closed group has no nontrivial abelian normal subgroup, then we show that it is a direct product of some (but not all) of these simple groups, and there are precisely [Formula: see text] examples. In the course of obtaining this classification, we develop a general framework for studying [Formula: see text]-closures of transitive permutation groups, which we hope will prove useful for investigating representations of finite groups as automorphism groups of graphs and digraphs, and in particular for attacking the long-standing polycirculant conjecture. In this direction, we apply our results, proving a dual to a 1939 theorem of Frucht from Algebraic Graph Theory. We also pose several open questions concerning closures of permutation groups.

Second Subgroup of General Linear Group in Dimention 2

In many practical situations we know that a primitive group contains a given permutation and we want to know which group it can be; in some other practical situations we know the group and would like to know if it contains a permutation of some given type. For example, a group G ≤ Sn is said to be non-synchronizing if it is contained in the automorphism group of a non-trivial primitive graph with complete core, that is, with clique number equal to chromatic number. When trying to check if some group is synchronizing, typically, we have only partial information about the graph but enough to say that it has an automorphism of some type, and the goal would be to have in hand a classification of the primitive groups containing permutations of that type. As an illustration of this, the key ingredient in some of the results in [2] was the observation that the primitive graph understudy has a 2-cycle automorphism and hence the automorphism group of the graph is the symmetric group. For many more examples of the importance of knowing the groups that contain permutations of a given type. This type of investigation is certainly very natural since it appears on the eve of group theory, with Jordan, Burnside, Marggraff, but the difficulty of the problem is well illustrated by the very slow progress throughout the twentieth century. Given the new tools available (chiefly the classification of finite simple groups), the topic seems to have new momentum. Let Sn denote the symmetric group on n points; a permutation g∈Sn is said to be imprimitive if there exists an imprimitive group contain in gg. An imprimitive group is said to beminimally imprimitiveifit contains no transitive proper subgroup. An imprimitive group G≤Sn is said to be maximally imprimitive if for all g Sn\G, the group〈g, G 〉 is primitive. The next result, whose proof isstraightforward, provides some alternative characterizations of imprimitive permutations. Now in this paper we discuse Presentation for imprimitive second Subgroup of general linear group in dimention 2 over the field of pk-elements .

Surge: a fast open-source chemical graph generator

Chemical structure generators are used in cheminformatics to produce or enumerate virtual molecules based on a set of boundary conditions. The result can then be tested for properties of interest, such as adherence to measured data or for their suitability as drugs. The starting point can be a potentially fuzzy set of fragments or a molecular formula. In the latter case, the generator produces the set of constitutional isomers of the given input formula. Here we present the novel constitutional isomer generator surge based on the canonical generation path method. Surge uses the nauty package to compute automorphism groups of graphs. We outline the working principles of surge and present benchmarking results which show that surge is currently the fastest structure generator. Surge is available under a liberal open-source license.

A New Family of Algebraically Defined Graphs with Small Automorphism Group

Let $p$ be an odd prime, $q=p^e$, $e \geq 1$, and $\mathbb{F} = \mathbb{F}_q$ denote the finite field of $q$ elements. Let $f: \mathbb{F}^2\to \mathbb{F}$ and $g: \mathbb{F}^3\to \mathbb{F}$ be functions, and let $P$ and $L$ be two copies of the 3-dimensional vector space $\mathbb{F}^3$. Consider a bipartite graph $\Gamma_\mathbb{F} (f, g)$ with vertex partitions $P$ and $L$ and with edges defined as follows: for every $(p)=(p_1,p_2,p_3)\in P$ and every $[l]= [l_1,l_2,l_3]\in L$, $\{(p), [l]\} = (p)[l]$ is an edge in $\Gamma_\mathbb{F} (f, g)$ if $$p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = g(p_1,p_2,l_1).$$The following question appeared in Nassau: Given $\Gamma_\mathbb{F} (f, g)$, is it always possible to find a function $h:\mathbb{F}^2\to \mathbb{F}$ such that the graph $\Gamma_\mathbb{F} (f, h)$ with the same vertex set as $\Gamma_\mathbb{F} (f, g)$ and with edges $(p)[l]$ defined in a similar way by the system $$p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = h(p_1,l_1),$$ is isomorphic to $\Gamma_\mathbb{F} (f, g)$ for infinitely many $q$? In this paper we show that the answer to the question is negative and the graphs $\Gamma_{\mathbb{F}_p}(p_1\ell_1, p_1\ell_1p_2(p_1 + p_2 + p_1p_2))$ provide such an example for $p \equiv 1 \pmod{3}$. Our argument is based on proving that the automorphism group of these graphs has order $p$, which is the smallest possible order of the automorphism group of graphs of the form $\Gamma_{\mathbb{F}}(f, g)$.

Wreath product in automorphism groups of graphs

AbstractThe automorphism group of the composition of graphs contains the wreath product of the automorphism groups of the corresponding graphs. The classical problem considered by Sabidussi and Hemminger was under what conditions has no other automorphisms. In this paper we consider questions related to the converse: if the automorphism group of a graph is a wreath product , are the smaller groups necessarily automorphism groups of graphs? And if so, are the corresponding smaller graphs involved in the construction? We consider these questions for the wreath product in its natural imprimitive action (which refers to the results by Sabidussi and Hemminger), and in generalization to colored graphs, which seems to be a more appropriate setting. For this case we have a fairly complete answer. Yet, we also consider the same problems for the wreath product in its product action. This turns out to be more complicated and we have only partial results. Our considerations in this part lead to interesting open questions involving hypergraphs and to an analogue of the Sabidussi–Hemminger problem for a related graph construction.

FINITE TWO-DISTANCE-TRANSITIVE DIHEDRANTS

AbstractA noncomplete graph is $2$ -distance-transitive if, for $i \in \{1,2\}$ and for any two vertex pairs $(u_1,v_1)$ and $(u_2,v_2)$ with the same distance i in the graph, there exists an element of the graph automorphism group that maps $(u_1,v_1)$ to $(u_2,v_2)$ . This paper determines the family of $2$ -distance-transitive Cayley graphs over dihedral groups, and it is shown that if the girth of such a graph is not $4$ , then either it is a known $2$ -arc-transitive graph or it is isomorphic to one of the following two graphs: $ {\mathrm {K}}_{x[y]}$ , where $x\geq 3,y\geq 2$ , and $G(2,p,({p-1})/{4})$ , where p is a prime and $p \equiv 1 \ (\operatorname {mod}\, 8)$ . Then, as an application of the above result, a complete classification is achieved of the family of $2$ -geodesic-transitive Cayley graphs for dihedral groups.

A 2-arc Transitive Hexavalent Nonnormal Cayley Graph on A119

A Cayley graph Γ=Cay(G,S) is said to be normal if the base group G is normal in AutΓ. The concept of the normality of Cayley graphs was first proposed by M.Y. Xu in 1998 and it plays a vital role in determining the full automorphism groups of Cayley graphs. In this paper, we construct an example of a 2-arc transitive hexavalent nonnormal Cayley graph on the alternating group A119. Furthermore, we determine the full automorphism group of this graph and show that it is isomorphic to A120.

Quantum symmetry of graph $C^{\ast }$-algebras at\cr critical inverse temperature

We give a notion of quantum automorphism group of graph C*-algebras without sink at critical inverse temperature. This is defined to be the universal object of a category of CQG's having a linear action in the sense of [11] and preserving the KMS state at critical inverse temperature. We show that this category for a certain KMS state at critical inverse temperature coincides with the category introduced in [11] for a class of graphs. We also introduce an orthogonal filtration on Cuntz algebra with respect to the unique KMS state and show that the category of CQG's preserving the orthogonal filtration coincides with the category introduced in this paper.

Generalized Gardiner–Praeger graphs and their symmetries

A subgroup of the automorphism group of a graph acts half-arc-transitively on the graph if it acts transitively on the vertex-set and on the edge-set of the graph but not on the arc-set of the graph. If the full automorphism group of the graph acts half-arc-transitively, the graph is said to be half-arc-transitive.In 1994 Gardiner and Praeger introduced two families of tetravalent arc-transitive graphs, called the C±1 and the C±ε graphs, that play a prominent role in the characterization of the tetravalent graphs admitting an arc-transitive group of automorphisms with a normal elementary abelian subgroup such that the corresponding quotient graph is a cycle. All of the Gardiner–Praeger graphs are arc-transitive but admit a half-arc-transitive group of automorphisms. Quite recently, Potočnik and Wilson introduced the family of CPM graphs, which are generalizations of the Gardiner–Praeger graphs. Most of these graphs are arc-transitive, but some of them are half-arc-transitive. In fact, at least up to order 1000, each tetravalent half-arc-transitive loosely-attached graph of odd radius having vertex-stabilizers of order greater than 2 is isomorphic to a CPM graph.In this paper we determine the automorphism group of the CPM graphs and investigate isomorphisms between them. Moreover, we determine which of these graphs are 2-arc-transitive, which are arc-transitive but not 2-arc-transitive, and which are half-arc-transitive.

Partition and Colored Distances in Graphs Induced to Subsets of Vertices and Some of Its Applications

If G is a graph and P is a partition of V(G), then the partition distance of G is the sum of the distances between all pairs of vertices that lie in the same part of P. A colored distance is the dual concept of the partition distance. These notions are motivated by a problem in the facility location network and applied to several well-known distance-based graph invariants. In this paper, we apply an extended cut method to induce the partition and color distances to some subsets of vertices which are not necessary a partition of V(G). Then, we define a two-dimensional weighted graph and an operator to prove that the induced partition and colored distances of a graph can be obtained from the weighted Wiener index of a two-dimensional weighted quotient graph induced by the transitive closure of the Djoković–Winkler relation as well as by any partition that is coarser. Finally, we utilize our main results to find some upper bounds for the modified Wiener index and the number of orbits of partial cube graphs under the action of automorphism group of graphs.

Quantum automorphisms of folded cube graphs

We show that the quantum automorphism group of the Clebsch graph is $SO_5^{-1}$. This answers a question by Banica, Bichon and Collins from 2007. More general for odd $n$, the quantum automorphism group of the folded $n$-cube graph is $SO_n^{-1}$. Furthermore, we show that if the automorphism group of a graph contains a pair of disjoint automorphisms this graph has quantum symmetry.

An Improved Isomorphism Test for Bounded-tree-width Graphs

We give a new FPT algorithm testing isomorphism of n -vertex graphs of tree-width k in time 2 kpolylog(k) n 3 , improving the FPT algorithm due to Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh (FOCS 2014), which runs in time 2 O(k5 log k) n 5 . Based on an improved version of the isomorphism-invariant graph decomposition technique introduced by Lokshtanov et al., we prove restrictions on the structure of the automorphism groups of graphs of tree-width k . Our algorithm then makes heavy use of the group theoretic techniques introduced by Luks (JCSS 1982) in his isomorphism test for bounded degree graphs and Babai (STOC 2016) in his quasipolynomial isomorphism test. In fact, we even use Babai’s algorithm as a black box in one place. We also give a second algorithm that, at the price of a slightly worse running time 2 O(k2 log k) n 3 , avoids the use of Babai’s algorithm and, more importantly, has the additional benefit that it can also be used as a canonization algorithm.

Nonlocal games and quantum permutation groups

We present a strong connection between quantum information and the theory of quantum permutation groups. Specifically, we define a notion of quantum isomorphisms of graphs based on quantum automorphisms from the theory of quantum groups, and then show that this is equivalent to the previously defined notion of quantum isomorphism corresponding to perfect quantum strategies to the isomorphism game. Moreover, we show that two connected graphs X and Y are quantum isomorphic if and only if there exists x∈V(X) and y∈V(Y) that are in the same orbit of the quantum automorphism group of the disjoint union of X and Y. This connection links quantum groups to the more concrete notion of nonlocal games and physically observable quantum behaviours. In this work, we exploit this by using ideas and results from quantum information in order to prove new results about quantum automorphism groups of graphs, and about quantum permutation groups more generally. In particular, we show that asymptotically almost surely all graphs have trivial quantum automorphism group. Furthermore, we use examples of quantum isomorphic graphs from previous work to construct an infinite family of graphs which are quantum vertex transitive but fail to be vertex transitive, answering a question from the quantum permutation group literature.Our main tool for proving these results is the introduction of orbits and orbitals (orbits on ordered pairs) of quantum permutation groups. We show that the orbitals of a quantum permutation group form a coherent configuration/algebra, a notion from the field of algebraic graph theory. We then prove that the elements of this quantum orbital algebra are exactly the matrices that commute with the magic unitary defining the quantum group. We furthermore show that quantum isomorphic graphs admit an isomorphism of their quantum orbital algebras which maps the adjacency matrix of one graph to that of the other.We hope that this work will encourage new collaborations among the communities of quantum information, quantum groups, and algebraic graph theory.

Twisted Lawrence-Krammer representations

Lawrence-Krammer representations are an important family of linear representations of Artin-Tits groups of small type, which are known, under some assumptions on the parameters, to be faithful when the type is spherical (or more generally when they are restricted to the Artin-Tits monoid) and irreducible when the type is connected.Here, we investigate an analogue of these representations — introduced by Digne in the spherical cases — for every Artin-Tits monoid that appears as the submonoid of fixed points of an Artin-Tits monoid of small type under a group of graph automorphisms, and for the corresponding Artin-Tits group.Under the same assumptions on the parameters as in the small type cases, we first show that these so-called “twisted Lawrence-Krammer representations” are faithful, and we then prove, by computing their formulas when the group of graph automorphisms is of order two or three, their irreducibility in all the spherical and connected cases but one.

Partial difference sets and amorphic Cayley schemes in non‐abelian 2‐groups

AbstractIn this paper, we consider regular automorphism groups of graphs in the RT2 family and the Davis‐Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of non‐abelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT2 family and Davis‐Xiang family and their amorphic abelian Cayley schemes to produce amorphic non‐abelian Cayley schemes.