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Register a new userOn the automorphism groups of graphs with twice prime valency
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A graph is edge-transitive if its automorphism group acts transitively on the edge set. In this paper, we investigate the automorphism groups of edge-transitive graphs of odd order and twice prime valency. Let $Gamma$ be a connected graph of odd order and twice prime valency, and let $G$ be a subgroup of the automorphism group of $Ga$. In the case where $G$ acts transitively on the edges and quasiprimitively on the vertices of $Ga$, we prove that either $G$ is almost simple or $G$ is a primitive group of affine type. If further $G$ is an almost simple primitive group then, with two exceptions, the socle of $G$ acts transitively on the edges of $Gamma$.
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In 2011, Fang et al. in (J. Combin. Theory A 118 (2011) 1039-1051) posed the following problem: Classify non-normal locally primitive Cayley graphs of finite simple groups of valency $d$, where either $dleq 20$ or $d$ is a prime number. The only case for which the complete solution of this problem is known is of $d=3$. Except this, a lot of efforts have been made to attack this problem by considering the following problem: Characterize finite nonabelian simple groups which admit non-normal locally primitive Cayley graphs of certain valency $dgeq4$. Even for this problem, it was only solved for the cases when either $dleq 5$ or $d=7$ and the vertex stabilizer is solvable. In this paper, we make crucial progress towards the above problems by completely solving the second problem for the case when $dgeq 11$ is a prime and the vertex stabilizer is solvable.
An automorphism of a graph is called quasi-semiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malniv{c}, Mart{i}nez and Maruv{s}iv{c} in 2013, as a generalization of the well-known semiregular automorphism of a graph. Symmetric graphs of valency three or four, admitting a quasi-semiregular automorphism, have been classified in recent two papers. Let $pgeq 5$ be a prime and $Gamma$ a connected symmetric graph of valency $p$ admitting a quasi-semiregular automorphism. In this paper, we first prove that either $Gamma$ is a connected Cayley graph $rm{Cay}(M,S)$ such that $M$ is a $2$-group admitting a fixed-point-free automorphism of order $p$ with $S$ as an orbit of involutions, or $Gamma$ is a normal $N$-cover of a $T$-arc-transitive graph of valency $p$ admitting a quasi-semiregular automorphism, where $T$ is a non-abelian simple group and $N$ is a nilpotent group. Then in case $p=5$, we give a complete classification of such graphs $Gamma$ such that either $rm{Aut}(Gamma)$ has a solvable arc-transitive subgroup or $Gamma$ is $T$-arc-transitive with $T$ a non-abelian simple group. We also construct the first infinite family of symmetric graphs that have a quasi-semiregular automorphism and an insolvable full automorphism group.
We explore graph theoretical properties of minimal prime graphs of finite solvable groups. In finite group theory studying the prime graph of a group has been an important topic for the past almost half century. Recently prime graphs of solvable groups have been characterized in graph theoretical terms only. This now allows the study of these graphs with methods from graph theory only. Minimal prime graphs turn out to be of particular interest, and in this paper we pursue this further by exploring, among other things, diameters, Hamiltonian cycles and the property of being self-complementary for minimal prime graphs. We also study a new, but closely related notion of minimality for prime graphs and look into counting minimal prime graphs.
A graph $Ga=(V,E)$ is called a Cayley graph of some group $T$ if the automorphism group $Aut(Ga)$ contains a subgroup $T$ which acts on regularly on $V$. If the subgroup $T$ is normal in $Aut(Ga)$ then $Ga$ is called a normal Cayley graph of $T$. Let $r$ be an odd prime. Fang et al. cite{FMW} proved that, with a finite number of exceptions for finite simple group $T$, every connected symmetric Cayley graph of $T$ of valency $r$ is normal. In this paper, employing maximal factorizations of finite almost simple groups, we work out a possible list of those exceptions for $T$.
Comparability graphs are graphs which have transitive orientations. The dimension of a poset is the least number of linear orders whose intersection gives this poset. The dimension ${rm dim}(X)$ of a comparability graph $X$ is the dimension of any transitive orientation of X, and by $k$-DIM we denote the class of comparability graphs $X$ with ${rm dim}(X) le k$. It is known that the complements of comparability graphs are exactly function graphs and permutation graphs equal 2-DIM. In this paper, we characterize the automorphism groups of permutation graphs similarly to Jordans characterization for trees (1869). For permutation graphs, there is an extra operation, so there are some extra groups not realized by trees. For $k ge 4$, we show that every finite group can be realized as the automorphism group of some graph in $k$-DIM, and testing graph isomorphism for $k$-DIM is GI-complete.
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