scholarly journals Some Properties of Coprime Graph of Dihedral Group D_2n When n is a Prime Power

2020 ◽  
Vol 3 (1) ◽  
pp. 34-38
Author(s):  
Abdul Gazir S ◽  
I Gede Adhitya Wisnu Wardhana ◽  
Ni Wayan Switrayni ◽  
Qurratul Aini
Keyword(s):  
Bipartite Graph ◽  
Prime Power ◽  
Dihedral Group ◽  
Complete Bipartite Graph ◽  
Algebraic Structures ◽  
Multipartite Graph

The Study of algebraic structures, especially on graphs theory, leads to anew topics of research in recent years. In this paper, the algebraic structures that will be represented by a coprime graph are the dihedral group and its subgroups. The coprime graph of a group G, denoted by \Gamma_D_2n is a graph whose vertices are elements of G and two distinct vertices a and b are adjacent if only if (|a,|b|)=1. Some properties of the coprime graph of a dihedral group D_2n are obtained. One of the results is if n is prime then \Gamma_D_2n is a complete bipartite graph. Moreover, if n is the power of prime then \Gamma_D_2n is a multipartite graph.

LOCALLY PRIMITIVE GRAPHS OF PRIME-POWER ORDER

2009 ◽  
Vol 86 (1) ◽  
pp. 111-122 ◽  
Author(s):  
CAI HENG LI ◽  
JIANGMIN PAN ◽  
LI MA
Keyword(s):  
Bipartite Graph ◽  
Cayley Graph ◽  
Complete Graph ◽  
Prime Power ◽  
Complete Bipartite Graph ◽  
Prime Power Order ◽  
Normal Cayley Graph ◽  
Normal Cover ◽  
Vertex Transitive

AbstractLet Γ be a finite connected undirected vertex transitive locally primitive graph of prime-power order. It is shown that either Γ is a normal Cayley graph of a 2-group, or Γ is a normal cover of a complete graph, a complete bipartite graph, or Σ×l, where Σ=Kpm with p prime or Σ is the Schläfli graph (of order 27). In particular, either Γ is a Cayley graph, or Γ is a normal cover of a complete bipartite graph.

scholarly journals TETRAVALENT s-TRANSITIVE GRAPHS OF ORDER TWICE A PRIME POWER

2010 ◽  
Vol 88 (2) ◽  
pp. 277-288 ◽  
Author(s):  
JIN-XIN ZHOU ◽  
YAN-QUAN FENG
Keyword(s):  
Automorphism Group ◽  
Bipartite Graph ◽  
Projective Geometry ◽  
Prime Power ◽  
Complete Bipartite Graph ◽  
Three Dimensional ◽  
Transitive Graph ◽  
Incidence Graph ◽  
Normal Cover ◽  
Transitive Graphs

AbstractA graph is s-transitive if its automorphism group acts transitively on s-arcs but not on (s+1)-arcs in the graph. Let X be a connected tetravalent s-transitive graph of order twice a prime power. In this paper it is shown that s=1,2,3 or 4. Furthermore, if s=2, then X is a normal cover of one of the following graphs: the 4-cube, the complete graph of order 5, the complete bipartite graph K5,5 minus a 1-factor, or K7,7 minus a point-hyperplane incidence graph of the three-dimensional projective geometry PG(2,2); if s=3, then X is a normal cover of the complete bipartite graph of order 4; if s=4, then X is a normal cover of the point-hyperplane incidence graph of the three-dimensional projective geometry PG(2,3). As an application, we classify the tetravalent s-transitive graphs of order 2p2 for prime p.

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 $n$.

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 .

scholarly journals An Exact Turán Result for Tripartite 3-Graphs

10.37236/5203 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Adam Sanitt ◽  
John Talbot
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Maximum Size ◽  
Unique Graph

Mantel's theorem says that among all triangle-free graphs of a given order the balanced complete bipartite graph is the unique graph of maximum size. We prove an analogue of this result for 3-graphs. Let $K_4^-=\{123,124,134\}$, $F_6=\{123,124,345,156\}$ and $\mathcal{F}=\{K_4^-,F_6\}$: for $n\neq 5$ the unique $\mathcal{F}$-free 3-graph of order $n$ and maximum size is the balanced complete tripartite 3-graph $S_3(n)$ (for $n=5$ it is $C_5^{(3)}=\{123,234,345,145,125\}$). This extends an old result of Bollobás that $S_3(n) $ is the unique 3-graph of maximum size with no copy of $K_4^-=\{123,124,134\}$ or $F_5=\{123,124,345\}$.

Sign in / Sign up

Export Citation Format

Share Document