Let $\Gamma$ be a connected $7$-valent symmetric Cayley graph on a finite non-abelian simple group $G$. If $\Gamma$ is not normal, Li {\em et al.} [On 7-valent symmetric Cayley graphs âŠ
Let $\Gamma$ be a connected $7$-valent symmetric Cayley graph on a finite non-abelian simple group $G$. If $\Gamma$ is not normal, Li {\em et al.} [On 7-valent symmetric Cayley graphs of finite simple groups, J. Algebraic Combin. 56 (2022) 1097-1118] characterised the group pairs $(\mathrm{soc}(\mathrm{Aut}(\Gamma)/K),GK/K)$, where $K$ is a maximal intransitive normal subgroup of $\mathrm{Aut}(\Gamma)$. In this paper, we improve this result by proving that if $\Gamma$ is not normal, then $\mathrm{Aut}(\Gamma)$ contains an arc-transitive non-abelian simple normal subgroup $T$ such that $G<T$ and $(T,G)=(\mathrm{A}_{n},\mathrm{A}_{n-1})$ with $n=7$, $3\cdot 7$, $3^2\cdot 7$, $2^2\cdot 3\cdot 7$, $2^3\cdot3\cdot7$, $2^3\cdot3^2\cdot5\cdot7$, $2^4\cdot3^2\cdot5\cdot7$, $2^6\cdot3\cdot7$, $2^7\cdot3\cdot7$, $2^6\cdot3^2\cdot7$, $2^6\cdot3^4\cdot5^2\cdot7$, $2^8\cdot3^4\cdot5^2\cdot7$, $2^7\cdot3^4\cdot5^2\cdot7$, $2^{10}\cdot3^2\cdot7$, $2^{24}\cdot3^2\cdot7$. Furthermore, $\mathrm{soc}(\mathrm{Aut}(\Gamma)/R)=(T\times R)/R$, where $R$ is the largest solvable normal subgroup of $\mathrm{Aut}(\Gamma)$.
In this paper we investigate the full automorphism groups of six-valent symmetric Cayley graphs Î = Cay (G,S) for finite non-abelian simple groups G. We prove that for most finite âŠ
In this paper we investigate the full automorphism groups of six-valent symmetric Cayley graphs Π= Cay (G,S) for finite non-abelian simple groups G. We prove that for most finite non-abelian simple groups G, if Πcontains no cycle of length 4, then Aut Π= G · Aut (G,S), where Aut (G,S) †S 6 .
We give a characterization of the automorphism groups of connected prime-valent symmetric Cayley graphs on finite (non-abelian) simple groups.
We give a characterization of the automorphism groups of connected prime-valent symmetric Cayley graphs on finite (non-abelian) simple groups.
A Cayley graph [Formula: see text] is said to be normal if [Formula: see text] is normal in [Formula: see text]. In this paper, we investigate the normality problem of âŠ
A Cayley graph [Formula: see text] is said to be normal if [Formula: see text] is normal in [Formula: see text]. In this paper, we investigate the normality problem of the connected 11-valent symmetric Cayley graphs [Formula: see text] of finite nonabelian simple groups [Formula: see text], where the vertex stabilizer [Formula: see text] is soluble for [Formula: see text] and [Formula: see text]. We prove that either [Formula: see text] is normal or [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. Further, 11-valent symmetric nonnormal Cayley graphs of [Formula: see text], [Formula: see text] and [Formula: see text] are constructed. This provides some more examples of nonnormal 11-valent symmetric Cayley graphs of finite nonabelian simple groups after the first graph of this kind (of valency 11) was constructed by Fang, Ma and Wang in 2011.
A Cayley graph $\Ga=\Cay(G,S)$ is said to be normal if the right-regular representation of $G$ is normal in $\Aut\Ga$. In this paper, we investigate the normality problem of the connected âŠ
A Cayley graph $\Ga=\Cay(G,S)$ is said to be normal if the right-regular representation of $G$ is normal in $\Aut\Ga$. In this paper, we investigate the normality problem of the connected 13-valent symmetric Cayley graphs $\Ga$ of finite nonabelian simple groups $G$, where the vertex stabilizer $\A_v$ is soluble for $\A=\Aut\Ga$ and $v\in V\Ga$. We prove that $\Ga$ is either normal or $G=\A_{12}$, $\A_{38}$, $\A_{116}$, $\A_{207}$, $\A_{311}$, $\A_{935}$ or $\A_{1871}$. Further, 13-valent symmetric non-normal Cayley graphs of $\A_{38}$, $\A_{116}$ and $\A_{207}$ are constructed. This provides some more examples of non-normal 13-valent symmetric Cayley graphs of finite nonabelian simple groups since such graph (of valency 13) was first constructed by Fang, Ma and Wang in (J. Comb. Theory A 118, 1039--1051, 2011).
Let $\Gamma$ be a bipartite graph, and let $\mathrm{Aut}\Gamma$ be the full automorphism group of the graph $\Gamma$. A subgroup $G\leqslant \mathrm{Aut}\Gamma$ is said to be bi-regular on $\Gamma$ if âŠ
Let $\Gamma$ be a bipartite graph, and let $\mathrm{Aut}\Gamma$ be the full automorphism group of the graph $\Gamma$. A subgroup $G\leqslant \mathrm{Aut}\Gamma$ is said to be bi-regular on $\Gamma$ if $G$ preserves the bipartition and acts regularly on both parts of $\Gamma$, while the graph $\Gamma$ is called a bi-Cayley graph of $G$ in this case. A subgroup $X\leqslant \mathrm{Aut} \Gamma$ is said to be bi-quasiprimitive on $\Gamma$ if the bipartition-preserving subgroup of $X$ is a quasiprimitive group on each part of $\Gamma$.
 In this paper, a characterization is given for the connected bi-Cayley graphs on nonabelian simple groups which have prime valency and admit bi-quasiprimitive groups.
Let $A$ be an abelian group and let $\iota$ be the automorphism of $A$ defined by $i:a\mapsto a^{-1}$. A Cayley graph $\Gamma=\mathrm{Cay}(A,S)$ is said to have an automorphism group \emph{as âŠ
Let $A$ be an abelian group and let $\iota$ be the automorphism of $A$ defined by $i:a\mapsto a^{-1}$. A Cayley graph $\Gamma=\mathrm{Cay}(A,S)$ is said to have an automorphism group \emph{as small as possible} if $\mathrm{Aut}(\Gamma)= A\rtimes\langle i\rangle$. In this paper, we show that almost all Cayley graphs on abelian groups have automorphism group as small as possible, proving a conjecture of Babai and Godsil.
Let $A$ be an abelian group and let $\iota$ be the automorphism of $A$ defined by $i:a\mapsto a^{-1}$. A Cayley graph $\Gamma=\mathrm{Cay}(A,S)$ is said to have an automorphism group \emph{as âŠ
Let $A$ be an abelian group and let $\iota$ be the automorphism of $A$ defined by $i:a\mapsto a^{-1}$. A Cayley graph $\Gamma=\mathrm{Cay}(A,S)$ is said to have an automorphism group \emph{as small as possible} if $\mathrm{Aut}(\Gamma)= A\rtimes\langle i\rangle$. In this paper, we show that almost all Cayley graphs on abelian groups have automorphism group as small as possible, proving a conjecture of Babai and Godsil.
A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group. In this paper we give âŠ
A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group. In this paper we give conditions for when a Cayley graph on an abelian group can be represented as a Cayley graph on a generalized dihedral group, and conditions for when the converse is true.
The present work is devoted to characterize the family of symmetric undirected Cayley graphs over finite Abelian groups for degrees 4 and 6.
The present work is devoted to characterize the family of symmetric undirected Cayley graphs over finite Abelian groups for degrees 4 and 6.
The present work is devoted to characterize the family of symmetric undirected Cayley graphs over finite Abelian groups for degrees 4 and 6.
The present work is devoted to characterize the family of symmetric undirected Cayley graphs over finite Abelian groups for degrees 4 and 6.
Let S n denote the symmetric group of degree n with n â„ 3, S = { c n = (1 2 ⯠n), [Formula: see text], (1 2)} and âŠ
Let S n denote the symmetric group of degree n with n â„ 3, S = { c n = (1 2 ⯠n), [Formula: see text], (1 2)} and Î n = Cay(S n , S) be the Cayley graph on S n with respect to S. In this paper, we show that Î n (n â„ 13) is a normal Cayley graph, and that the full automorphism group of Î n is equal to Aut(Î n ) = R(S n ) â ăInn(Ï) â S n à †2 , where R(S n ) is the right regular representation of S n , Ï = (1 2)(3 n)(4 nâ1)(5 nâ2) ⯠(â S n ), and Inn(Ï) is the inner isomorphism of S n induced by Ï.
Let $S_n$ denote the symmetric group of degree $n$ with $n\geq 3$. Set $S=\{c_n=(1\ 2\ldots \ n),c_n^{-1},(1\ 2)\}$. Let $\Gamma_n=\mathrm{Cay}(S_n,S)$ be the Cayley graph on $S_n$ with respect to $S$. âŠ
Let $S_n$ denote the symmetric group of degree $n$ with $n\geq 3$. Set $S=\{c_n=(1\ 2\ldots \ n),c_n^{-1},(1\ 2)\}$. Let $\Gamma_n=\mathrm{Cay}(S_n,S)$ be the Cayley graph on $S_n$ with respect to $S$. In this paper, we show that $\Gamma_n$ ($n\geq 13$) is a normal Cayley graph, and that the full automorphism group of $\Gamma_n$ is equal to $\mathrm{Aut}(\Gamma_n)=R(S_n)\rtimes \langle\mathrm{Inn}(\phi)\rangle\cong S_n\rtimes \mathbb{Z}_2$, where $R(S_n)$ is the right regular representation of $S_n$, $\phi=(1\ 2)(3\ n)(4\ n-1)(5\ n-2)\cdots$ $(\in S_n)$, and $\mathrm{Inn}(\phi)$ is the inner isomorphism of $S_n$ induced by $\phi$.
With the classification of the finite simple groups complete, much work has gone into the study of maximal subgroups of almost simple groups. In this volume the authors investigate the âŠ
With the classification of the finite simple groups complete, much work has gone into the study of maximal subgroups of almost simple groups. In this volume the authors investigate the maximal subgroups of the finite classical groups and present research into these groups as well as proving many new results. In particular, the authors develop a unified treatment of the theory of the 'geometric subgroups' of the classical groups, introduced by Aschbacher, and they answer the questions of maximality and conjugacy and obtain the precise shapes of these groups. Both authors are experts in the field and the book will be of considerable value not only to group theorists, but also to combinatorialists and geometers interested in these techniques and results. Graduate students will find it a very readable introduction to the topic and it will bring them to the very forefront of research in group theory.
Let G be a finite nonabelian simple group and let Î be a connected undirected Cayley graph for G. The possible structures for the full automorphism group AutÎ are specified. âŠ
Let G be a finite nonabelian simple group and let Î be a connected undirected Cayley graph for G. The possible structures for the full automorphism group AutÎ are specified. Then, for certain finite simple groups G, a sufficient condition is given under which G is a normal subgroup of AutÎ. Finally, as an application of these results, several new half-transitive graphs are constructed. Some of these involve the sporadic simple groups G = J1, J4, Ly and BM, while others fall into two infinite families and involve the Ree simple groups and alternating groups. The two infinite families contain examples of half-transitive graphs of arbitrarily large valency.
Abstract Let G be a group acting symmetrically on a graph ÎŁ, let G 1 be a subgroup of G minimal among those that act symmetrically on ÎŁ, and let âŠ
Abstract Let G be a group acting symmetrically on a graph ÎŁ, let G 1 be a subgroup of G minimal among those that act symmetrically on ÎŁ, and let G 2 be a subgroup of G 1 maximal among those normal subgroups of G 1 which contain no member except 1 which fixes a vertex of ÎŁ. The most precise result of this paper is that if ÎŁ has prime valency p , then either ÎŁ is a bipartite graph or G 2 acts regularly on ÎŁ or G 1 | G 2 is a simple group which acts symmetrically on a graph of valency p which can be constructed from ÎŁ and does not have more vertices than ÎŁ. The results on vertexâtransitive groups necessary to establish results like this are also included.
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. âŠ
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 G v the vertex stabilizer of a vertex v â V(X) in G. For the case p=3, the exact structure of G v has been determined by DjokoviÄ and Miller in [Regular groups of automorphisms of cubic graphs, J. Combin. Theory (Ser. B) 29 (1980) 195 â 230]. For the case p=5, all the possibilities of G v have been given by Guo and Feng in [A note on pentavalent s-transitive graphs, Discrete Math.312 (2012) 2214 â 2216]. In this paper, we deal with the case p=7 and determine the exact structure of the vertex stabilizer G v .
Let G be a finite non-abelian simple group and let Î be a connected tetravalent 2-arc-transitive G-regular graph. In 2004, Fang, Li, and Xu proved that either G is normal âŠ
Let G be a finite non-abelian simple group and let Î be a connected tetravalent 2-arc-transitive G-regular graph. In 2004, Fang, Li, and Xu proved that either G is normal in the full automorphism group Aut(Î) of Î, or G is one of up to 22 exceptional candidates. In this paper, the number of exceptions is reduced to 7, and for each one, it is shown that Aut(Î) has a normal arc-transitive non-abelian simple subgroup T such that Gâ€T and the pair (G, T) is explicitly given. Furthermore, there exists a G-regular (T,2)-arc-transitive graph for each of the 7 pairs (G, T).
We give a characterization of the automorphism groups of connected prime-valent symmetric Cayley graphs on finite (non-abelian) simple groups.
We give a characterization of the automorphism groups of connected prime-valent symmetric Cayley graphs on finite (non-abelian) simple groups.
A plenty of contributions have been done on symmetric Cayley graphs on nonabelian simple groups, but the only known complete classification of such graphs with composite valency is of valency âŠ
A plenty of contributions have been done on symmetric Cayley graphs on nonabelian simple groups, but the only known complete classification of such graphs with composite valency is of valency 4 (provided 2-arc-transitivity) by Fang et al. [Europ. J. Combin. 25 (2004), 1107â1116] and Du and Feng [Comm. Algebra 47 (2019), 4565â4574]. This naturally motivates this work for classifying 2-arc-transitive hexavalent Cayley graphs on nonabelian simple groups. It is proved that these graphs are either normal or (An,2)-arc-transitive Cayley graphs on Anâ1 where n is among 11 specific numbers dividing 27·33·53. A specific example is also constructed.
Surfaces of general type with canonical map of degree d bigger than 8 have bounded geometric genus and irregularity. In particular the irregularity is at most 2 if dâ„10. In âŠ
Surfaces of general type with canonical map of degree d bigger than 8 have bounded geometric genus and irregularity. In particular the irregularity is at most 2 if dâ„10. In the present paper, the existence of surfaces with d = 10 and all possible irregularities, surfaces with d = 12 and irregularity 1 and 2, and surfaces with d = 14 and irregularity 0 and 1 is proven, by constructing these surfaces as Z23-covers of certain rational surfaces. These results together with the construction by C. Rito of a surface with d = 12 and irregularity 0 show that all the possibilities for the irregularity in the cases d = 10, d = 12 can occur, whilst the existence of a surface with d = 14 and irregularity 2 is still an open problem.