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.

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 The K -Size Edge Metric Dimension of Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Tanveer Iqbal ◽  
Muhammad Naeem Azhar ◽  
Syed Ahtsham Ul Haq Bokhary
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Complete Bipartite Graph ◽  
Petersen Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Generalized Petersen Graph

In this paper, a new concept k -size edge resolving set for a connected graph G in the context of resolvability of graphs is defined. Some properties and realizable results on k -size edge resolvability of graphs are studied. The existence of this new parameter in different graphs is investigated, and the k -size edge metric dimension of path, cycle, and complete bipartite graph is computed. It is shown that these families have unbounded k -size edge metric dimension. Furthermore, the k-size edge metric dimension of the graphs Pm □ Pn, Pm □ Cn for m, n ≥ 3 and the generalized Petersen graph is determined. It is shown that these families of graphs have constant k -size edge metric dimension.

Arc-Transitive Graphs of Square-free Order with Valency 11

2021 ◽  
Vol 28 (04) ◽  
pp. 645-654
Author(s):  
Guang Li ◽  
Bo Ling ◽  
Zaiping Lu
Keyword(s):  
Simple Group ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Cayley Graphs ◽  
Complete List ◽  
Dihedral Groups ◽  
Transitive Graphs ◽  
Free Order ◽  
Normal Cayley Graphs

In this paper, we present a complete list of connected arc-transitive graphs of square-free order with valency 11. The list includes the complete bipartite graph [Formula: see text], the normal Cayley graphs of dihedral groups and the graphs associated with the simple group [Formula: see text] and [Formula: see text], where [Formula: see text] is a prime.

scholarly journals Invariant subspaces of matrix groups and elementary-abelian covers of K4,4

Filomat ◽  
2011 ◽  
Vol 25 (4) ◽  
pp. 37-53
Author(s):  
Bostjan Kuzman
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Invariant Subspaces ◽  
Matrix Groups ◽  
Abelian Covers ◽  
Groups Of Automorphisms ◽  
Transitive Subgroup

We study lifting conditions for groups of automorphisms of the complete bipartite graph K4,4. In particular, for p?2 a prime we construct, up to isomorphism of projections, all minimal p-elementary abelian covers of K4,4 such that the respective covering projections admit a lift of some arc-transitive subgroup of Aut(K4,4).

A neigborhood union condition for nonadjacent vertices in graphs

2017 ◽  
Vol 09 (05) ◽  
pp. 1750060
Author(s):  
Farzad Shaveisi
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Simple Graph ◽  
Strongly Regular Graphs ◽  
Star Graph ◽  
Regular Graphs ◽  
Strongly Regular ◽  
Union Condition ◽  
Bounded Graph

A simple graph [Formula: see text] is called [Formula: see text]-bounded if for every two nonadjacent vertices [Formula: see text] of [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text], where [Formula: see text] denotes the set of neighbors of the vertex [Formula: see text] in [Formula: see text]. In this paper, some properties of [Formula: see text]-bounded graphs are studied. It is shown that any bipartite [Formula: see text]-bounded graph is a complete bipartite graph with at most two horns; in particular, any [Formula: see text]-bounded tree is either a star or a two-star graph. Also, we prove that any non-end vertex of every [Formula: see text]-bounded graph is contained in either a triangle or a rectangle. Among other results, it is shown that all regular [Formula: see text]-bounded graphs are strongly regular graphs. Finally, we determine that how many edges can an [Formula: see text]-bounded graph have?

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

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 $n$.

Generalization of the Symmetric Starter of the Orthogonal Double Covers of the Complete Bipartite Graph

2006 ◽  
Vol 16 (2) ◽  
pp. 171-177
Author(s):  
S. A. El-Serafi ◽  
R. A. El-Shanawany ◽  
M. Sh. Higazy
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Double Covers

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 .

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

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