On the Stabilizer of the Automorphism Group of a 4-valent Vertex-transitive Graph with Odd-prime-power Order

2003 ◽  
Vol 19 (1) ◽  
pp. 83-86 ◽  
Author(s):  
Yan-quan Feng ◽  
Jin Ho Kwak ◽  
Ming-yao Xu
Keyword(s):  
Automorphism Group ◽  
Prime Power ◽  
Prime Power Order ◽  
Transitive Graph ◽  
Vertex Transitive ◽  
Vertex Transitive Graph

scholarly journals Semiregular elements in cubic vertex-transitive graphs and the restricted Burnside problem

2014 ◽  
Vol 157 (1) ◽  
pp. 45-61
Author(s):  
PABLO SPIGA
Keyword(s):  
Automorphism Group ◽  
Positive Solution ◽  
Cayley Graph ◽  
Burnside Problem ◽  
Transitive Graph ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Restricted Burnside Problem ◽  
Vertex Transitive Graph ◽  
Transitive Graphs

AbstractIn this paper, we prove that the maximal order of a semiregular element in the automorphism group of a cubic vertex-transitive graph Γ does not tend to infinity as the number of vertices of Γ tends to infinity. This gives a solution (in the negative) to a conjecture of Peter Cameron, John Sheehan and the author [4, conjecture 2].However, with an application of the positive solution of the restricted Burnside problem, we show that this conjecture holds true when Γ is either a Cayley graph or an arc-transitive graph.

scholarly journals Absolutely split metacyclic groups and weak metacirculants

2019 ◽  
Vol 18 (06) ◽  
pp. 1950117 ◽  
Author(s):  
Li Cui ◽  
Jin-Xin Zhou
Keyword(s):  
Automorphism Group ◽  
Prime Power ◽  
Necessary Condition ◽  
Prime Power Order ◽  
Metacyclic Groups ◽  
Vertex Transitive ◽  
Open Question ◽  
Positive Integers ◽  
The Relationship ◽  
Sufficient And Necessary Condition

Let [Formula: see text] be positive integers, and let [Formula: see text] be a split metacyclic group such that [Formula: see text]. We say that [Formula: see text] is absolutely split with respect to[Formula: see text] provided that for any [Formula: see text], if [Formula: see text], then there exists [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, we give a sufficient and necessary condition for the group [Formula: see text] being absolutely split. This generalizes a result of Sanming Zhou and the second author in [Weak metacirculants of odd prime power order, J. Comb. Theory A 155 (2018) 225–243]. We also use this result to investigate the relationship between metacirculants and weak metacirculants. Metacirculants were introduced by Alspach and Parsons in [Formula: see text] and have been a rich source of various topics since then. As a generalization of this class of graphs, Marušič and Šparl in 2008 introduced the so-called weak metacirculants. A graph is called a weak metacirculant if it has a vertex-transitive metacyclic automorphism group. In this paper, it is proved that a weak metacirculant of [Formula: see text]-power order is a metacirculant if and only if it has a vertex-transitive split metacyclic automorphism group. This provides a partial answer to an open question in the literature.

scholarly journals Cubic Vertex-Transitive Non-Cayley Graphs of Order $8p$

10.37236/2087 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Jin-Xin Zhou ◽  
Yan-Quan Feng
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Infinite Family ◽  
The Other ◽  
Transitive Graph ◽  
Vertex Transitive ◽  
Unique Graph ◽  
Vertex Transitive Graph

A graph is vertex-transitive if its automorphism group acts transitively on its vertices. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper, the cubic vertex-transitive non-Cayley graphs of order $8p$ are classified for each prime $p$. It follows from this classification that there are two sporadic and two infinite families of such graphs, of which the sporadic ones have order $56$,  one infinite family exists for every prime $p>3$ and the other family exists if and only if $p\equiv 1\mod 4$. For each family there is a unique graph for a given order.

Local Expansion of Symmetrical Graphs

1992 ◽  
Vol 1 (1) ◽  
pp. 1-11 ◽  
Author(s):  
László Babai ◽  
Mario Szegedy
Keyword(s):  
Automorphism Group ◽  
Finite Subset ◽  
Harmonic Mean ◽  
Expansion Rate ◽  
Transitive Graph ◽  
Local Expansion ◽  
Vertex Set ◽  
Vertex Transitive ◽  
Edge Transitive ◽  
Vertex Transitive Graph

A graph is vertex-transitive (edge-transitive) if its automorphism group acts transitively on the vertices (edges, resp.). The expansion rate of a subset S of the vertex set is the quotient e(S):= |∂(S)|/|S|, where ∂(S) denotes the set of vertices not in S but adjacent to some vertex in S. Improving and extending previous results of Aldous and Babai, we give very simple proofs of the following results. Let X be a (finite or infinite) vertex-transitive graph and let S be a finite subset of the vertices. If X is finite, we also assume |S| ≤|V(X)/2. Let d be the diameter of S in the metric induced by X. Then e(S) ≥1/(d + 1); and e(S) ≥ 2/(d +2) if X is finite and d is less than the diameter of X. If X is edge-transitive then |δ(S)|/|S| ≥ r/(2d), where ∂(S) denotes the set of edges joining S to its complement and r is the harmonic mean of the minimum and maximum degrees of X. – Diverse applications of the results are mentioned.

scholarly journals Regular graphs and stability

1975 ◽  
Vol 20 (3) ◽  
pp. 377-384 ◽  
Author(s):  
D. A. Holton ◽  
Douglas D. Grant
Keyword(s):  
Automorphism Group ◽  
Regular Graphs ◽  
Transitive Graph ◽  
Trivial Vertex ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Vertex Transitive Graph ◽  
Stable Vertex ◽  
Transitive Graphs

AbstractWe show that a graph G is semi-stable at vertex v if and only if the set of vertices of G adjacent to v is fixed by the automorphism group of Gv, the subgraph of G obtained by deleting v and its incident edges. This result leads to a neat proof that regular graphs are semi-stable at each vertex. We then investigate stable regular graphs, concentrating mainly on stable vertex-transitive graphs. We conjecture that if G is a non-trivial vertex-transitive graph, then G is stable if and only if γ(G) contains a transposition, offering some evidence for its truth.

scholarly journals Vertex-Transitive Digraphs of Order $p^5$ are Hamiltonian

10.37236/4034 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Jun-Yang Zhang
Keyword(s):  
Automorphism Group ◽  
Prime Power ◽  
Prime Power Order ◽  
Connected Digraph ◽  
Vertex Transitive ◽  
Transitive Subgroup ◽  
Connected Vertex

We prove that connected vertex-transitive digraphs of order $p^{5}$ (where $p$ is a prime) are Hamiltonian, and a connected digraph whose automorphism group contains a finite vertex-transitive subgroup $G$ of prime power order such that $G'$ is generated by two elements or elementary abelian is Hamiltonian.

scholarly journals Cayley Graphs Without a Bounded Eigenbasis

2020 ◽  
Author(s):  
Ashwin Sah ◽  
Mehtaab Sawhney ◽  
Yufei Zhao
Keyword(s):  
Cayley Graph ◽  
Orthonormal Basis ◽  
Cayley Graphs ◽  
The Other ◽  
Transitive Graph ◽  
Classic Result ◽  
Other Hand ◽  
Small Set ◽  
Vertex Transitive ◽  
Vertex Transitive Graph

Abstract Does every $n$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $O(1/\sqrt{n})$? While the answer is yes for abelian groups, we show that it is no in general. On the other hand, we show that every $n$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $O(\sqrt{\log n / n})$, and that this bound is nearly best possible. Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon–Roichman. His proof relies on the existence of a bounded eigenbasis for abelian Cayley graphs, which we now know cannot hold for general groups. On the other hand, we navigate around this obstruction and extend Naor’s result to nonabelian groups.

scholarly journals Automorphism orbits of finite groups

1986 ◽  
Vol 40 (2) ◽  
pp. 253-260 ◽  
Author(s):  
Thomas J. Laffey ◽  
Desmond MacHale
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Groups ◽  
Prime Power ◽  
Structure Theorem ◽  
Equivalence Classes ◽  
Prime Power Order ◽  
A Finite Group

AbstractLet G be a finite group and let Aut(G) be its automorphism group. Then G is called a k-orbit group if G has k orbits (equivalence classes) under the action of Aut(G). (For g, hG, we have g ~ h if ga = h for some Aut(G).) It is shown that if G is a k-orbit group, then kGp + 1, where p is the least prime dividing the order of G. The 3-orbit groups which are not of prime-power order are classified. It is shown that A5 is the only insoluble 4-orbit group, and a structure theorem is proved about soluble 4-orbit groups.

scholarly journals Two-Distance-Primitive Graphs

10.37236/8890 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Wei Jin ◽  
Ci Xuan Wu ◽  
Jin Xin Zhou
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Complete Classification ◽  
Second Step ◽  
Transitive Graph ◽  
Vertex Transitive ◽  
Vertex Transitive Graph ◽  
Cyclic Graph ◽  
Acts 2

A 2-distance-primitive graph is a vertex-transitive graph whose vertex stabilizer is primitive on both the first step and the second step neighborhoods. Let $\Gamma$ be such a graph. This paper shows that either $\Gamma$ is a cyclic graph, or $\Gamma$ is a complete bipartite graph, or $\Gamma$ has girth at most $4$ and the vertex stabilizer acts faithfully on both the first step and the second step neighborhoods. Also a complete classification is given of such graphs  satisfying that the vertex stabilizer acts $2$-transitively on the second step neighborhood. Finally, we determine the unique 2-distance-primitive graph which is  locally cyclic.

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