Vertex-primitive half-transitive graphs

AbstractGiven an infinite family of finite primitive groups, conditions are found which ensure that almost all the orbitals are not self-paired. If p is a prime number congruent to ±1(mod 10), these conditions apply to the groups P S L (2, p) acting on the cosets of a subgroup isomorphic to A5. In this way, infinitely many vertex-primitive ½-transitive graphs which are not metacirculants are obtained.

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Properties of Classes of Random Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548300001346 ◽

1994 ◽

Vol 3 (4) ◽

pp. 435-454 ◽

Cited By ~ 1

Author(s):

Neal Brand ◽

Steve Jackson

Keyword(s):

Random Graphs ◽

Higher Order ◽

Order Property ◽

First Order ◽

New Class ◽

Vertex Transitive ◽

Vertex Transitive Graphs ◽

Almost All ◽

Transitive Graphs

In [11] it is shown that the theory of almost all graphs is first-order complete. Furthermore, in [3] a collection of first-order axioms are given from which any first-order property or its negation can be deduced. Here we show that almost all Steinhaus graphs satisfy the axioms of almost all graphs and conclude that a first-order property is true for almost all graphs if and only if it is true for almost all Steinhaus graphs. We also show that certain classes of subgraphs of vertex transitive graphs are first-order complete. Finally, we give a new class of higher-order axioms from which it follows that large subgraphs of specified type exist in almost all graphs.

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A Family of Half-Transitive Graphs

The Electronic Journal of Combinatorics ◽

10.37236/3140 ◽

2013 ◽

Vol 20 (1) ◽

Author(s):

Jing Chen ◽

Cai Heng Li ◽

Ákos Seress

Keyword(s):

Cayley Graphs ◽

Infinite Family ◽

Transitive Graphs

We construct an infinite family of half-transitive graphs, which contains infinitely many Cayley graphs, and infinitely many non-Cayley graphs.

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Affine Primitive Groups and Semisymmetric Graphs

The Electronic Journal of Combinatorics ◽

10.37236/2549 ◽

2013 ◽

Vol 20 (2) ◽

Cited By ~ 1

Author(s):

Hua Han ◽

Zaiping Lu

Keyword(s):

Bipartite Graph ◽

Complete Bipartite Graph ◽

Infinite Family ◽

Permutation Groups ◽

Primitive Groups ◽

The Third ◽

Semisymmetric Graphs

In this paper, we investigate semisymmetric graphs which arise from affine primitive permutation groups. We give a characterization of such graphs, and then construct an infinite family of semisymmetric graphs which contains the Gray graph as the third smallest member. Then, as a consequence, we obtain a factorization,of the complete bipartite graph $K_{p^{sp^t},p^{sp^t}}$ into connected semisymmetric graphs, where $p$ is an prime, $1\le t\le s$ with $s\ge2$ while $p=2$.

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Primes in short intervals

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004196001193 ◽

1997 ◽

Vol 122 (2) ◽

pp. 193-205 ◽

Cited By ~ 2

Author(s):

HONGZE LI

Keyword(s):

Prime Number ◽

Short Intervals ◽

Almost All

In 1982, Glyn Harman [2] proved that for almost all n, the interval [n, n+n(1/10)+ε] contains a prime number. By this we mean that the set of n[les ]N for which the interval does not contain a prime has measure o(N) as n→+∞. It follows from Huxley's work [6] that if θ>1/6 then there will almost always be asymptotically nθ(log n)−1 primes in the interval [n, n+nθ]. In 1983, Glyn Harman [3] pointed that for almost all n, the interval [n, n+n(1/12)+ε] contains a prime number, and meantime Heath-Brown gave the outline of this result in [5]. The exponent was reduced to 1/13 by Jia [10], 2/27 by Li [12] and 1/14 by Jia [11], and meantime N. Watt [16] got the same result. In this paper we shall prove the following result.THEOREM. For almost all n, the intervalformula herecontains a prime number.

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Zassenhaus conjecture on torsion units holds for SL(2, 𝑝) and SL(2, 𝑝2)

Journal of Group Theory ◽

10.1515/jgth-2018-0113 ◽

2019 ◽

Vol 22 (5) ◽

pp. 953-974

Author(s):

Ángel del Río ◽

Mariano Serrano

Keyword(s):

Finite Group ◽

Prime Number ◽

Infinite Family ◽

Solvable Groups ◽

Integral Group ◽

Rational Group ◽

Torsion Unit ◽

Zassenhaus Conjecture ◽

A Finite Group ◽

Torsion Units

Abstract H. J. Zassenhaus conjectured that any unit of finite order and augmentation 1 in the integral group ring {\mathbb{Z}G} of a finite group G is conjugate in the rational group algebra {\mathbb{Q}G} to an element of G. We prove the Zassenhaus conjecture for the groups {\mathrm{SL}(2,p)} and {\mathrm{SL}(2,p^{2})} with p a prime number. This is the first infinite family of non-solvable groups for which the Zassenhaus conjecture has been proved. We also prove that if {G=\mathrm{SL}(2,p^{f})} , with f arbitrary and u is a torsion unit of {\mathbb{Z}G} with augmentation 1 and order coprime with p, then u is conjugate in {\mathbb{Q}G} to an element of G. By known results, this reduces the proof of the Zassenhaus conjecture for these groups to proving that every unit of {\mathbb{Z}G} of order a multiple of p and augmentation 1 has order actually equal to p.

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The finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge 4$

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-01-02768-4 ◽

2001 ◽

Vol 353 (9) ◽

pp. 3511-3529 ◽

Cited By ~ 43

Author(s):

Cai Heng Li

Keyword(s):

Transitive Graphs ◽

Vertex Primitive

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SELF-COMPLEMENTARY VERTEX-TRANSITIVE GRAPHS NEED NOT BE CAYLEY GRAPHS

Bulletin of the London Mathematical Society ◽

10.1112/s0024609301008505 ◽

2001 ◽

Vol 33 (6) ◽

pp. 653-661 ◽

Cited By ~ 13

Author(s):

CAI HENG LI ◽

CHERYL E. PRAEGER

Keyword(s):

Wreath Product ◽

Cayley Graphs ◽

Infinite Family ◽

Permutation Groups ◽

General Study ◽

Product Action ◽

Vertex Transitive ◽

Vertex Transitive Graphs ◽

Transitive Graphs

A construction is given of an infinite family of finite self-complementary, vertex-transitive graphs which are not Cayley graphs. To the authors' knowledge, these are the first known examples of such graphs. The nature of the construction was suggested by a general study of the structure of self-complementary, vertex-transitive graphs. It involves the product action of a wreath product of permutation groups.

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Quadratic approximation in ℚp

International Journal of Number Theory ◽

10.1142/s1793042115500128 ◽

2014 ◽

Vol 11 (01) ◽

pp. 193-209 ◽

Cited By ~ 3

Author(s):

Yann Bugeaud ◽

Tomislav Pejković

Keyword(s):

Prime Number ◽

Continued Fractions ◽

Quadratic Approximation ◽

Almost All

Let p be a prime number. Let w2 and [Formula: see text] denote the exponents of approximation defined by Mahler and Koksma, respectively, in their classifications of p-adic numbers. It is well-known that every p-adic number ξ satisfies [Formula: see text], with [Formula: see text] for almost all ξ. By means of Schneider's continued fractions, we give explicit examples of p-adic numbers ξ for which the function [Formula: see text] takes any prescribed value in the interval (0, 1].

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