scholarly journals Random Cyclic Triangle-Free Graphs of Prime Order

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Yu Jiang ◽  
Meilian Liang ◽  
Yanmei Teng ◽  
Xiaodong Xu
Keyword(s):  
Lower Bounds ◽  
Prime Order ◽  
Free Graph ◽  
Cyclic Analog ◽  
Integer Order ◽  
Free Process ◽  
Novel Approach ◽  
Cyclic Graphs ◽  
Composite Integer ◽  
Cyclic Graph

Cyclic triangle-free process (CTFP) is the cyclic analog of the triangle-free process. It begins with an empty graph of order n and generates a cyclic graph of order n by iteratively adding parameters, chosen uniformly at random, subject to the constraint that no triangle is formed in the cyclic graph obtained, until no more parameters can be added. The structure of a cyclic triangle-free graph of the prime order is different from that of composite integer order. Cyclic graphs of prime order have better properties than those of composite number order, which enables generating cyclic triangle-free graphs more efficiently. In this paper, a novel approach to generating cyclic triangle-free graphs of prime order is proposed. Based on the cyclic graphs of prime order, obtained by the CTFP and its variant, many new lower bounds on R 3 , t are computed, including R 3,34 ≥ 230 , R 3,35 ≥ 242 , R 3,36 ≥ 252 , R 3,37 ≥ 264 , R 3,38 ≥ 272 . Our experimental results demonstrate that all those related best known lower bounds, except the bound on R 3,34 , are improved by 5 or more.

Bounds on hyper-Wiener index of graphs

2017 ◽  
Vol 10 (03) ◽  
pp. 1750057
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Sadegh Rahimi
Keyword(s):  
Lower Bounds ◽  
Organic Molecules ◽  
Wiener Index ◽  
Upper And Lower Bounds ◽  
Pharmacological Properties ◽  
Graph Theoretic ◽  
Acyclic Graphs ◽  
Wiener Number ◽  
Number Formula ◽  
Cyclic Graphs

The Wiener number [Formula: see text] of a graph [Formula: see text] was introduced by Harold Wiener in connection with the modeling of various physic-chemical, biological and pharmacological properties of organic molecules in chemistry. Milan Randić introduced a modification of the Wiener index for trees (acyclic graphs), and it is known as the hyper-Wiener index. Then Klein et al. generalized Randić’s definition for all connected (cyclic) graphs, as a generalization of the Wiener index, denoted by [Formula: see text] and defined as [Formula: see text]. In this paper, we establish some upper and lower bounds for [Formula: see text], in terms of other graph-theoretic parameters. Moreover, we compute hyper-Wiener number of some classes of graphs.

scholarly journals Lower Bounds for the Size of Random Maximal $H$-Free Graphs

10.37236/93 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Guy Wolfovitz
Keyword(s):  
Lower Bound ◽  
Random Process ◽  
Lower Bounds ◽  
Random Permutation ◽  
Bipartite Graphs ◽  
Expected Number ◽  
Free Graph ◽  
Vertex Graph

We consider the next random process for generating a maximal $H$-free graph: Given a fixed graph $H$ and an integer $n$, start by taking a uniformly random permutation of the edges of the complete $n$-vertex graph $K_n$. Then, traverse the edges of $K_n$ according to the order imposed by the permutation and add each traversed edge to an (initially empty) evolving $n$-vertex graph - unless its addition creates a copy of $H$. The result of this process is a maximal $H$-free graph ${\Bbb M}_n(H)$. Our main result is a new lower bound on the expected number of edges in ${\Bbb M}_n(H)$, for $H$ that is regular, strictly $2$-balanced. As a corollary, we obtain new lower bounds for Turán numbers of complete, balanced bipartite graphs. Namely, for fixed $r \ge 5$, we show that ex$(n, K_{r,r}) = \Omega(n^{2-2/(r+1)}(\ln\ln n)^{1/(r^2-1)})$. This improves an old lower bound of Erdős and Spencer. Our result relies on giving a non-trivial lower bound on the probability that a given edge is included in ${\Bbb M}_n(H)$, conditioned on the event that the edge is traversed relatively (but not trivially) early during the process.

scholarly journals NON-CYCLIC GRAPH ASSOCIATED WITH A GROUP

2009 ◽  
Vol 08 (02) ◽  
pp. 243-257 ◽  
Author(s):  
A. ABDOLLAHI ◽  
A. MOHAMMADI HASSANABADI
Keyword(s):  
Solvable Group ◽  
Cyclic Group ◽  
Cyclic Subgroup ◽  
Maximum Size ◽  
Clique Number ◽  
Simple Graph ◽  
Complete Subgraph ◽  
Vertex Set ◽  
Cyclic Graphs ◽  
Cyclic Graph

We associate a graph [Formula: see text] to a non locally cyclic group G (called the non-cyclic graph of G) as follows: take G\ Cyc (G) as vertex set, where Cyc (G) = {x ∈ G | 〈x,y〉 is cyclic for all y ∈ G} is called the cyclicizer of G, and join two vertices if they do not generate a cyclic subgroup. For a simple graph Γ, w(Γ) denotes the clique number of Γ, which is the maximum size (if it exists) of a complete subgraph of Γ. In this paper we characterize groups whose non-cyclic graphs have clique numbers at most 4. We prove that a non-cyclic group G is solvable whenever [Formula: see text] and the equality for a non-solvable group G holds if and only if G/ Cyc (G) ≅ A5 or S5.

scholarly journals A Generalization of Generalized Paley Graphs and New Lower Bounds for $R(3,q)$

10.37236/474 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Kang Wu ◽  
Wenlong Su ◽  
Haipeng Luo ◽  
Xiaodong Xu
Keyword(s):  
Finite Fields ◽  
Lower Bounds ◽  
Independent Set ◽  
Maximum Independent Set ◽  
Ramsey Numbers ◽  
Cyclic Graphs ◽  
Paley Graphs

Generalized Paley graphs are cyclic graphs constructed from quadratic or higher residues of finite fields. Using this type of cyclic graphs to study the lower bounds for classical Ramsey numbers, has high computing efficiency in both looking for parameter sets and computing clique numbers. We have found a new generalization of generalized Paley graphs, i.e. automorphism cyclic graphs, also having the same advantages. In this paper we study the properties of the parameter sets of automorphism cyclic graphs, and develop an algorithm to compute the order of the maximum independent set, based on which we get new lower bounds for $8$ classical Ramsey numbers: $R(3,22) \geq 131$, $R(3,23) \geq 137$, $R(3,25) \geq 154$, $R(3,28) \geq 173$, $R(3,29) \geq 184$, $R(3,30) \geq 190$, $R(3,31) \geq 199$, $R(3,32) \geq 214$. Furthermore, we also get $R(5,23) \geq 521$ based on $R(3,22) \geq 131$. These nine results above improve their corresponding best known lower bounds.

scholarly journals Nitrogen-doped porous carbon monoliths from polyacrylonitrile (PAN) and carbon nanotubes as electrodes for supercapacitors

2017 ◽  
Vol 7 (1) ◽  
Author(s):  
Yanqing Wang ◽  
Bunshi Fugetsu ◽  
Zhipeng Wang ◽  
Wei Gong ◽  
Ichiro Sakata ◽  
...  
Keyword(s):  
Carbon Nanotubes ◽  
Three Dimensional ◽  
Acid Base ◽  
Nitrogen Doped ◽  
Free Process ◽  
Novel Approach ◽  
Activated Carbon Monoliths ◽  
Template Free ◽  
Porous Activated Carbon ◽  
Electrodes For Supercapacitors

Abstract Nitrogen-doped porous activated carbon monoliths (NDP-ACMs) have long been the most desirable materials for supercapacitors. Unique to the conventional template based Lewis acid/base activation methods, herein, we report on a simple yet practicable novel approach to production of the three-dimensional NDP-ACMs (3D-NDP-ACMs). Polyacrylonitrile (PAN) contained carbon nanotubes (CNTs), being pre-dispersed into a tubular level of dispersions, were used as the starting material and the 3D-NDP-ACMs were obtained via a template-free process. First, a continuous mesoporous PAN/CNT based 3D monolith was established by using a template-free temperature-induced phase separation (TTPS). Second, a nitrogen-doped 3D-ACM with a surface area of 613.8 m2/g and a pore volume 0.366 cm3/g was obtained. A typical supercapacitor with our 3D-NDP-ACMs as the functioning electrodes gave a specific capacitance stabilized at 216 F/g even after 3000 cycles, demonstrating the advantageous performance of the PAN/CNT based 3D-NDP-ACMs.

scholarly journals Characteristic Graph vs Benzenoid Graph

2019 ◽  
Vol 7 (2) ◽  
pp. 135-147
Author(s):  
Sufia Aziz
Keyword(s):  
Electron Energy ◽  
Structural Property ◽  
Benzenoid Graph ◽  
Pi Index ◽  
Cyclic Graphs ◽  
Cyclic Graph

A characteristic graph is a tree representative of its corresponding benzenoid (cyclic) graph. It may contain necessary information of several properties of benzenoids. The PI-index of benzenoids and their characteristic graphs are compared by correlating it to a structural property (π-electron energy) of the benzenoids using MLR analysis. PI index being applicable to both trees and cyclic graphs, yielded required results for benzenoid and their characteristic graph.

scholarly journals Clumsy Packings of Graphs

10.37236/7942 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Maria Axenovich ◽  
Anika Kaufmann ◽  
Raphael Yuster
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Minimum Size ◽  
Maximum Cardinality ◽  
Natural Parameter ◽  
Free Graph ◽  
Minimum Cardinality ◽  
Empty Graph ◽  
Edge Disjoint ◽  
Hexagonal Grids

Let $G$ and $H$ be graphs. We say that $P$ is an $H$-packing of $G$ if $P$ is a set of edge-disjoint copies of $H$ in $G$. An $H$-packing $P$ is maximal if there is no other $H$-packing of $G$ that properly contains P. Packings of maximum cardinality have been studied intensively, with several recent breakthrough results. Here, we consider minimum cardinality maximal packings. An $H$-packing $P$ is called clumsy if it is maximal of minimum size. Let $\mathrm{cl}(G,H)$ be the size of a clumsy $H$-packing of $G$. We provide nontrivial bounds for $\mathrm{cl}(G,H)$, and in many cases asymptotically determine $\mathrm{cl}(G,H)$ for some generic classes of graphs G such as $K_n$ (the complete graph), $Q_n$ (the cube graph), as well as square, triangular, and hexagonal grids. We asymptotically determine $\mathrm{cl}(K_n,H)$ for every fixed non-empty graph $H$. In particular, we prove that  $$\mathrm{cl}(K_n, H) = \frac{\binom{n}{2}- \mathrm{ex}(n,H)}{|E(H)|}+o(\mathrm{ex}(n,H)),$$where $ex(n,H)$ is the extremal number of $H$. A related natural parameter is $\mathrm{cov}(G,H)$, that is the smallest number of copies of $H$ in $G$ (not necessarily edge-disjoint) whose removal from $G$ results in an $H$-free graph. While clearly $\mathrm{cov}(G,H) \leqslant\mathrm{cl}(G,H)$, all of our lower bounds for $\mathrm{cl}(G,H)$ apply to $\mathrm{cov}(G,H)$ as well.

scholarly journals New Computational Upper Bounds for Ramsey Numbers $R(3,k)$

10.37236/2824 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Jan Goedgebeur ◽  
Stanisław P. Radziszowski
Keyword(s):  
Lower Bounds ◽  
Independent Sets ◽  
Upper Bounds ◽  
Computational Techniques ◽  
Free Graph ◽  
Ramsey Numbers ◽  
Critical Graph ◽  
Minimum Number

Using computational techniques we derive six new upper bounds on the classical two-color Ramsey numbers: $R(3,10) \le 42$, $R(3,11) \le 50$, $R(3,13) \le 68$, $R(3,14) \le 77$, $R(3,15) \le 87$, and $R(3,16) \le 98$. All of them are improvements by one over the previously best known bounds. Let $e(3,k,n)$ denote the minimum number of edges in any triangle-free graph on $n$ vertices without independent sets of order $k$. The new upper bounds on $R(3,k)$ are obtained by completing the computation of the exact values of $e(3,k,n)$ for all $n$ with $k \leq 9$ and for all $n \leq 33$ for $k = 10$, and by establishing new lower bounds on $e(3,k,n)$ for most of the open cases for $10 \le k \le 15$. The enumeration of all graphs witnessing the values of $e(3,k,n)$ is completed for all cases with $k \le 9$. We prove that the known critical graph for $R(3,9)$ on 35 vertices is unique up to isomorphism. For the case of $R(3,10)$, first we establish that $R(3,10)=43$ if and only if $e(3,10,42)=189$, or equivalently, that if $R(3,10)=43$ then every critical graph is regular of degree 9. Then, using computations, we disprove the existence of the latter, and thus show that $R(3,10) \le 42$.

scholarly journals Spanning Simplicial Complexes of Uni-Cyclic Graphs

2015 ◽  
Vol 22 (04) ◽  
pp. 707-710 ◽  
Author(s):  
Imran Anwar ◽  
Zahid Raza ◽  
Agha Kashif
Keyword(s):  
Simplicial Complex ◽  
Connected Graph ◽  
Spanning Trees ◽  
Hilbert Series ◽  
Simplicial Complexes ◽  
Cyclic Graphs ◽  
Cyclic Graph

In this paper, we introduce the concept of the spanning simplicial complex Δs(G) associated to a simple finite connected graph G. We characterize all spanning trees of the uni-cyclic graph Un,m. In particular, we give a formula for computing the Hilbert series and h-vector of the Stanley-Reisner ring k[Δs(Un,m)]. Finally, we prove that the spanning simplicial complex Δs(Un,m) is shifted and hence is shellable.

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