scholarly journals Pentavalent Symmetric Graphs of Order $12p$

10.37236/720 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Song-Tao Guo ◽  
Jin-Xin Zhou ◽  
Yan-Quan Feng
Keyword(s):  
Automorphism Group ◽  
Complete Classification ◽  
Symmetric Graph ◽  
Symmetric Graphs

A graph is said to be symmetric if its automorphism group acts transitively on its arcs. In this paper, a complete classification of connected pentavalent symmetric graphs of order $12p$ is given for each prime $p$. As a result, a connected pentavalent symmetric graph of order $12p$ exists if and only if $p=2$, $3$, $5$ or $11$, and up to isomorphism, there are only nine such graphs: one for each $p=2$, $3$ and $5$, and six for $p=11$.

scholarly journals Cubic symmetric graphs of order twice an odd prime-power

2006 ◽  
Vol 81 (2) ◽  
pp. 153-164 ◽  
Author(s):  
Yan-Quan Feng ◽  
Jin Ho Kwak
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Prime Power ◽  
Symmetric Graph ◽  
Cubic Graphs ◽  
Full Automorphism Group ◽  
Symmetric Graphs ◽  
Regular Subgroup

AbstractAn automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. For a connected cubic symmetric graph X of order 2pn for an odd prime p, we show that if p ≠ 5, 7 then every Sylow p-subgroup of the full automorphism group Aut(X) of X is normal, and if p ≠3 then every s-regular subgroup of Aut(X) having a normal Sylow p-subgroup contains an (s − 1)-regular subgroup for each 1 ≦ s ≦ 5. As an application, we show that every connected cubic symmetric graph of order 2pn is a Cayley graph if p > 5 and we classify the s-regular cubic graphs of order 2p2 for each 1≦ s≦ 5 and each prime p. as a continuation of the authors' classification of 1-regular cubic graphs of order 2p2. The same classification of those of order 2p is also done.

scholarly journals Automorphisms of surfaces of general type with acting trivially in cohomology

2013 ◽  
Vol 149 (10) ◽  
pp. 1667-1684 ◽  
Author(s):  
Jin-Xing Cai ◽  
Wenfei Liu ◽  
Lei Zhang
Keyword(s):  
Automorphism Group ◽  
Minimal Surfaces ◽  
General Type ◽  
Cohomology Ring ◽  
Complete Classification ◽  
Surfaces Of General Type

AbstractIn this paper we prove that surfaces of general type with irregularity $q\geq 3$ are rationally cohomologically rigidified, and so are minimal surfaces $S$ with $q(S)= 2$ unless ${ K}_{S}^{2} = 8\chi ({ \mathcal{O} }_{S} )$. Here a surface $S$ is said to be rationally cohomologically rigidified if its automorphism group $\mathrm{Aut} (S)$ acts faithfully on the cohomology ring ${H}^{\ast } (S, \mathbb{Q} )$. As examples we give a complete classification of surfaces isogenous to a product with $q(S)= 2$ that are not rationally cohomologically rigidified.

scholarly journals THE VELDKAMP SPACE OF THE SMALLEST SLIM DENSE NEAR HEXAGON

2012 ◽  
Vol 10 (02) ◽  
pp. 1250082 ◽  
Author(s):  
R. M. GREEN ◽  
METOD SANIGA
Keyword(s):  
Black Holes ◽  
Quantum Information ◽  
Automorphism Group ◽  
Complete Classification ◽  
Recent Emergence ◽  
Different Types ◽  
Geometrical Concepts ◽  
Stabilizer Group ◽  
Near Hexagon

We give a detailed description of the Veldkamp space of the smallest slim dense near hexagon. This space is isomorphic to PG(7,2) and its 28- 1 = 255 Veldkamp points (that is, geometric hyperplanes of the near hexagon) fall into five distinct classes, each of which is uniquely characterized by the number of points/lines as well as by a sequence of the cardinalities of points of given orders and/or that of (grid-)quads of given types. For each type we also give its weight, stabilizer group within the full automorphism group of the near hexagon and the total number of copies. The totality of (255 choose 2)/3 = 10,795 Veldkamp lines split into 41 different types. We give a complete classification of them in terms of the properties of their cores (i.e. subconfigurations of points and lines common to all the three hyperplanes comprising a given Veldkamp line) and the types of the hyperplanes they are composed of. These findings may lend themselves into important physical applications, especially in view of recent emergence of a variety of closely related finite geometrical concepts linking quantum information with black holes.

scholarly journals Classification of Cubic Symmetric Tricirculants

10.37236/2371 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Istvan Kovacs ◽  
Klavdija Kutnar ◽  
Dragan Marusic ◽  
Steve Wilson
Keyword(s):  
Automorphism Group ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Equal Length ◽  
Symmetric Graph ◽  
Cycle Decomposition ◽  
Identity Automorphism

A tricirculant is a graph admitting a non-identity automorphism having three cycles of equal length in its cycle decomposition. A graph is said to be symmetric if its automorphism group acts transitively on the set of its arcs. In this paper it is shown that the complete bipartite graph $K_{3,3}$, the Pappus graph, Tutte's 8-cage and the unique cubic symmetric graph of order 54 are the only connected cubic symmetric tricirculants.

scholarly journals Two-geodesic-transitive graphs which are neighbor cubic or neighbor tetravalent

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2483-2488
Author(s):  
Wei Jin ◽  
Li Tan
Keyword(s):  
Automorphism Group ◽  
Complete Classification ◽  
Transitive Graphs

A vertex triple (u, v, w) with v adjacent to both u and w is called a 2-geodesic if u ? w and u,w are not adjacent. A graph ? is said to be 2-geodesic-transitive if its automorphism group is transitive on both arcs and 2-geodesics. In this paper, a complete classification of 2-geodesic-transitive graphs is given which are neighbor cubic or neighbor tetravalent.

Heptavalent Symmetric Graphs with Certain Conditions

2021 ◽  
Vol 28 (02) ◽  
pp. 243-252
Author(s):  
Jiali Du ◽  
Yanquan Feng ◽  
Yuqin Liu
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Simple Group ◽  
Simple Groups ◽  
Symmetric Graph ◽  
Symmetric Graphs ◽  
Vertex Transitive

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.

scholarly journals An algebraic characterization of symmetric graphs with a prime number of vertices

1972 ◽  
Vol 7 (1) ◽  
pp. 131-134 ◽  
Author(s):  
J.L. Berggren
Keyword(s):  
Automorphism Group ◽  
Prime Number ◽  
Simple Proof ◽  
Algebraic Characterization ◽  
Symmetric Graph ◽  
Symmetric Graphs ◽  
Multiplicative Subgroup

A graph Γ is called symmetric if its automorphism group is transitive on its vertices and edges. Let p be an odd prime, Z(p) the field of integers modulo p, and Z*(p) = (a ∈ Z(p) | a ≠ 0}, the multiplicative subgroup of Z(p). This paper gives a simple proof of the equivalence of two statements:(1) Γ is a symmetric graph with p vertices, each having degree n ≥ 1;(2) the integer n is an even divisor of p − 1 and Γ is isomorphic to the graph whose vertices are the elements of Z(p) and whose edges are the pairs {a, a+h} where a ∈ Z(p) and h ∈ H, the unique subgroup of Z*(p) of order n.In addition, the automorphism group of Γ is determined.

scholarly journals Multiplicative automatic sequences

2021 ◽  
Author(s):  
Jakub Konieczny ◽  
Mariusz Lemańczyk ◽  
Clemens Müllner
Keyword(s):  
Complete Classification ◽  
Automatic Sequences ◽  
Complex Valued

AbstractWe obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.

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