scholarly journals A Formula for the Energy of Circulant Graphs with Two Generators

2016 ◽  
Vol 2016 ◽  
pp. 1-4
Author(s):  
Justine Louis
Keyword(s):  
Complete Graph ◽  
Hamilton Cycle ◽  
Circulant Graphs

We derive closed formulas for the energy of circulant graphs generated by1andγ, whereγ⩾2is an integer. We also find a formula for the energy of the complete graph without a Hamilton cycle.

Symmetric Hamilton cycle decompositions of the complete graph

2003 ◽  
Vol 12 (1) ◽  
pp. 39-45 ◽  
Author(s):  
Jin Akiyama ◽  
Midori Kobayashi ◽  
Gisaku Nakamura
Keyword(s):  
Complete Graph ◽  
Hamilton Cycle ◽  
Cycle Decompositions

scholarly journals Perfect 1-Factorisations of Circulants with Small Degree

10.37236/2264 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Sarada Herke ◽  
Barbara Maenhaut
Keyword(s):  
Large Family ◽  
Small Degree ◽  
Hamilton Cycle ◽  
Perfect Matchings ◽  
Existence Problem ◽  
Circulant Graphs ◽  
Edge Disjoint ◽  
Even Order

A $1$-factorisation of a graph $G$ is a decomposition of $G$ into edge-disjoint $1$-factors (perfect matchings), and a perfect $1$-factorisation is a $1$-factorisation in which the union of any two of the $1$-factors is a Hamilton cycle.  We consider the problem of the existence of perfect $1$-factorisations of even order circulant graphs with small degree.  In particular, we characterise the $3$-regular circulant graphs that admit a perfect $1$-factorisation and we solve the existence problem for a large family of $4$-regular circulants.  Results of computer searches for perfect  $1$-factorisations of $4$-regular circulant graphs of orders up to $30$ are provided and some problems are posed.

scholarly journals The Threshold Probability for Long Cycles

2016 ◽  
Vol 26 (2) ◽  
pp. 208-247 ◽  
Author(s):  
ROMAN GLEBOV ◽  
HUMBERTO NAVES ◽  
BENNY SUDAKOV
Keyword(s):  
Random Graph ◽  
Complete Graph ◽  
Minimum Degree ◽  
Hamilton Cycle ◽  
Threshold Probability ◽  
Spanning Subgraph ◽  
Classic Result ◽  
Long Cycles

For a given graph G of minimum degree at least k, let Gp denote the random spanning subgraph of G obtained by retaining each edge independently with probability p = p(k). We prove that if p ⩾ (logk + loglogk + ωk(1))/k, where ωk(1) is any function tending to infinity with k, then Gp asymptotically almost surely contains a cycle of length at least k + 1. When we take G to be the complete graph on k + 1 vertices, our theorem coincides with the classic result on the threshold probability for the existence of a Hamilton cycle in the binomial random graph.

scholarly journals Biased Positional Games and Small Hypergraphs with Large Covers

10.37236/794 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Michael Krivelevich ◽  
Tibor Szabó
Keyword(s):  
Complete Graph ◽  
Winning Strategy ◽  
Hamilton Cycle ◽  
Covering Number ◽  
New Approach ◽  
Positional Games

We prove that in the biased $(1:b)$ Hamiltonicity and $k$-connectivity Maker-Breaker games ($k>0$ is a constant), played on the edges of the complete graph $K_n$, Maker has a winning strategy for $b\le(\log 2-o(1))n/\log n$. Also, in the biased $(1:b)$ Avoider-Enforcer game played on $E(K_n)$, Enforcer can force Avoider to create a Hamilton cycle when $b\le (1-o(1))n/\log n$. These results are proved using a new approach, relying on the existence of hypergraphs with few edges and large covering number.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 512
Author(s):  
Maryam Baghipur ◽  
Modjtaba Ghorbani ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Distance Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Simple Connected Graph ◽  
Second Largest Eigenvalue ◽  
Reciprocal Distance Matrix

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

scholarly journals Undirected Complete Graph to Design New Public Key Cryptosystem

2021 ◽  
Vol 1897 (1) ◽  
pp. 012045
Author(s):  
Karrar Taher R. Aljamaly ◽  
Ruma Kareem K. Ajeena
Keyword(s):  
Complete Graph ◽  
Public Key ◽  
Public Key Cryptosystem ◽  
New Public

scholarly journals Turán theorems for unavoidable patterns

2021 ◽  
pp. 1-20
Author(s):  
ANTÓNIO GIRÃO ◽  
BHARGAV NARAYANAN
Keyword(s):  
Complete Graph ◽  
Blow Up ◽  
Ramsey Numbers ◽  
Order Of Magnitude ◽  
Transitive Tournament ◽  
Unavoidable Patterns

Abstract We prove Turán-type theorems for two related Ramsey problems raised by Bollobás and by Fox and Sudakov. First, for t ≥ 3, we show that any two-colouring of the complete graph on n vertices that is δ-far from being monochromatic contains an unavoidable t-colouring when δ ≫ n−1/t, where an unavoidable t-colouring is any two-colouring of a clique of order 2t in which one colour forms either a clique of order t or two disjoint cliques of order t. Next, for t ≥ 3, we show that any tournament on n vertices that is δ-far from being transitive contains an unavoidable t-tournament when δ ≫ n−1/[t/2], where an unavoidable t-tournament is the blow-up of a cyclic triangle obtained by replacing each vertex of the triangle by a transitive tournament of order t. Conditional on a well-known conjecture about bipartite Turán numbers, both our results are sharp up to implied constants and hence determine the order of magnitude of the corresponding off-diagonal Ramsey numbers.

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