scholarly journals Affine Primitive Groups and Semisymmetric Graphs

10.37236/2549 ◽  
2013 ◽  
Vol 20 (2) ◽  
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 $p=2$.

scholarly journals On Arc-Transitive Metacyclic Covers of Graphs with Order Twice a Prime

10.37236/7146 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Zhaohong Huang ◽  
Jiangmin Pan
Keyword(s):  
Bipartite Graph ◽  
Prime Order ◽  
Complete Bipartite Graph ◽  
Infinite Family ◽  
Petersen Graph ◽  
Regular Graphs ◽  
Dihedral Groups ◽  
Hadamard Design ◽  
Symmetric Graphs ◽  
Abelian Covers

Quite a lot of attention has been paid recently to the characterization and construction of edge- or arc-transitive abelian (mostly cyclic or elementary abelian) covers of symmetric graphs, but there are rare results for nonabelian covers since the voltage graph techniques are generally not easy to be used in this case. In this paper, we will classify certain metacyclic arc-transitive covers of all non-complete symmetric graphs with prime valency and twice a prime order $2p$ (involving the complete bipartite graph ${\sf K}_{p,p}$, the Petersen graph, the Heawood graph, the Hadamard design on $22$ points and an infinite family of prime-valent arc-regular graphs of dihedral groups). A few previous results are extended.

THE TWO-ARC-TRANSITIVE GRAPHS OF SQUARE-FREE ORDER ADMITTING ALTERNATING OR SYMMETRIC GROUPS

2017 ◽  
Vol 104 (1) ◽  
pp. 127-144
Author(s):  
GAI XIA WANG ◽  
ZAI PING LU
Keyword(s):  
Finite Group ◽  
Bipartite Graph ◽  
Complete Graph ◽  
Complete Bipartite Graph ◽  
Symmetric Groups ◽  
Transitive Graph ◽  
A Finite Group ◽  
Transitive Graphs ◽  
Free Order

Let $G$ be a finite group with $\mathsf{soc}(G)=\text{A}_{c}$ for $c\geq 5$. A characterization of the subgroups with square-free index in $G$ is given. Also, it is shown that a $(G,2)$-arc-transitive graph of square-free order is isomorphic to a complete graph, a complete bipartite graph with a matching deleted or one of $11$ other graphs.

scholarly journals On a stronger reconstruction notion for monoids and clones

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mike Behrisch ◽  
Edith Vargas-García
Keyword(s):  
Bipartite Graph ◽  
Random Graph ◽  
Ordered Set ◽  
Permutation Groups ◽  
Random Tournament ◽  
Automatic Homeomorphicity ◽  
Dense Group ◽  
Automatic Action ◽  
Transformation Monoids

Abstract Motivated by reconstruction results by Rubin, we introduce a new reconstruction notion for permutation groups, transformation monoids and clones, called automatic action compatibility, which entails automatic homeomorphicity. We further give a characterization of automatic homeomorphicity for transformation monoids on arbitrary carriers with a dense group of invertibles having automatic homeomorphicity. We then show how to lift automatic action compatibility from groups to monoids and from monoids to clones under fairly weak assumptions. We finally employ these theorems to get automatic action compatibility results for monoids and clones over several well-known countable structures, including the strictly ordered rationals, the directed and undirected version of the random graph, the random tournament and bipartite graph, the generic strictly ordered set, and the directed and undirected versions of the universal homogeneous Henson graphs.

scholarly journals Metric dimension and pattern avoidance in graphs

2018 ◽  
Author(s):  
Jesse Geneson
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Connected Graph ◽  
Complete Bipartite Graph ◽  
Pattern Avoidance ◽  
Metric Dimension ◽  
Maximum Value ◽  
Vertex Graph ◽  
Dimension 2

We prove that the maximum possible number of edges in a graph of diameter $D$ and edge metric dimension $k$ is at most $(\lfloor \frac{2D}{3}\rfloor +1)^{k}+k \sum_{i = 1}^{\lceil \frac{D}{3}\rceil } (2i-1)^{k-1}$, sharpening the bound of $\binom{k}{2}+k D^{k-1}+D^{k}$ from Zubrilina (2018). We also prove that there is no subgraph of diameter $D$ with more than $(D+1)^{k}$ vertices in any connected graph of metric dimension $k$, and there is no subgraph of diameter $D$ with more than $(D+1)^{k}$ edges in any connected graph of edge metric dimension $k$. In particular, we prove that the maximum value of $n$ such that a graph of metric dimension $\leq k$ can contain the complete graph $K_{n}$ as a subgraph is $n = 2^{k}$. We also prove that the maximum value of $n$ such that a graph of metric dimension or edge metric dimension $\leq k$ can contain the complete bipartite graph $K_{n,n}$ as a subgraph is $2^{\theta(k)}$. Furthermore, we show that the maximum value of $n$ such that a graph of edge metric dimension $\leq k$ can contain $K_{1,n}$ as a subgraph is $n = 2^{k}$. We also show that the maximum value of $n$ such that a graph of metric dimension $\leq k$ can contain $K_{1,n}$ as a subgraph is $3^{k}-O(k)$. In addition, we prove that the $d$-dimensional grid $\prod_{i = 1}^{d} P_{r_{i}}$ has edge metric dimension at most $d$. This generalizes two results of Kelenc et al (2016), that non-path grids have edge metric dimension $2$ and that $d$-dimensional hypercubes have edge metric dimension at most $d$. We also provide a characterization of $n$-vertex graphs with edge metric dimension $n-2$, answering a question of Zubrilina. As a result of this characterization, we prove that any connected $n$-vertex graph $G$ such that $edim(G) = n-2$ has diameter at most $5$. More generally, we prove that any connected $n$-vertex graph with edge metric dimension $n-k$ has diameter at most $3k-1$.

Graceful Labeling for Some Complete Bipartite Graph

2018 ◽  
Vol 9 (12) ◽  
pp. 2147-2152
Author(s):  
V. Raju ◽  
M. Paruvatha vathana
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Graceful Labeling

Gaunilo Parodies Anselm

2014 ◽  
Vol 17 (1) ◽  
pp. 45-71
Author(s):  
Geo Siegwart
Keyword(s):  
Detailed Comparison ◽  
The Third ◽  
Common Structure ◽  
Logical Reconstruction ◽  
Points Of View ◽  
And Function

The main objective is an interpretation of the island parody, in particular a logical reconstruction of the parodying argument that stays close to the text. The parodied reasoning is identified as the proof in the second chapter of the Proslogion, more specifically, this proof as it is represented by Gaunilo in the first chapter of his Liber pro insipiente. The second task is a detailed comparison between parodied and parodying argument as well as an account of their common structure. The third objective is a tentative characterization of the nature and function of parodies of arguments. It seems that parodying does not add new pertinent points of view to the usual criticism of an argument.

On the Crossing Number of $K_{m,n}$

10.37236/1748 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Nagi H. Nahas
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.

scholarly journals Join Products K2,3 + Cn

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 925
Author(s):  
Michal Staš
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number ◽  
Arithmetic Means ◽  
Minimum Number ◽  
Edge Crossings

The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .

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