TWO AUTOMATIC SPANNING TREES IN SMALL CANCELLATION GROUP PRESENTATIONS

1996 ◽  
Vol 06 (04) ◽  
pp. 429-440 ◽  
Author(s):  
KARIN JOHNSGARD
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Group Element ◽  
Geometric Form ◽  
Small Cancellation ◽  
Automatic Structure ◽  
Group Presentations ◽  
Cancellation Group ◽  
Small Cancellation Group

Gersten and Short showed that group presentations satisfying certain minimal small cancellation conditions are automatic. The automatic structure they described does not have unique group element representatives. It is easy, however, to modify their language to produce an automatic spanning tree of geodesics: that is, an automatic structure of geodesics which is both prefix-closed and has unique representatives. When combined with a construction showing that all geodesic representatives of a group element are encoded in a tiling of a particular geometric form, this shows that in fact the language of lexicographically least geodesics is also automatic (it is clearly a spanning tree).

EVOLUTION OF SHANGHAI STOCK MARKET BASED ON MAXIMAL SPANNING TREES

2012 ◽  
Vol 27 (03) ◽  
pp. 1350022 ◽  
Author(s):  
CHUNXIA YANG ◽  
YING SHEN ◽  
BINGYING XIA
Keyword(s):  
Stock Market ◽  
Stock Prices ◽  
Spanning Tree ◽  
Stock Price ◽  
Spanning Trees ◽  
Turning Points ◽  
Single Step ◽  
P Value ◽  
Power Law Distribution ◽  
Structure Variation

In this paper, using a moving window to scan through every stock price time series over a period from 2 January 2001 to 11 March 2011 and mutual information to measure the statistical interdependence between stock prices, we construct a corresponding weighted network for 501 Shanghai stocks in every given window. Next, we extract its maximal spanning tree and understand the structure variation of Shanghai stock market by analyzing the average path length, the influence of the center node and the p-value for every maximal spanning tree. A further analysis of the structure properties of maximal spanning trees over different periods of Shanghai stock market is carried out. All the obtained results indicate that the periods around 8 August 2005, 17 October 2007 and 25 December 2008 are turning points of Shanghai stock market, at turning points, the topology structure of the maximal spanning tree changes obviously: the degree of separation between nodes increases; the structure becomes looser; the influence of the center node gets smaller, and the degree distribution of the maximal spanning tree is no longer a power-law distribution. Lastly, we give an analysis of the variations of the single-step and multi-step survival ratios for all maximal spanning trees and find that two stocks are closely bonded and hard to be broken in a short term, on the contrary, no pair of stocks remains closely bonded for a long time.

The Causal Tracing Method of Fast Moving Consumer Goods Logistics Quality Based on Extended Spanning Tree

2013 ◽  
Vol 694-697 ◽  
pp. 3480-3483
Author(s):  
Shou Wen Ji ◽  
Zeng Rong Su ◽  
Zhi Hua Zhang
Keyword(s):  
Consumer Goods ◽  
Fast Moving ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Logic Model ◽  
Fast Moving Consumer Goods ◽  
Tracing Algorithm ◽  
Tracing Method ◽  
Logistics Quality

The paper analyzes the extended spanning trees elements corresponding to fast-moving consumer goods (FMCG) logistics quality. According to extended spanning tree, we establish a logic model of FMCGs logistics quality causal tracing. At last, the paper gives out tracing algorithm and specific tracing process of FMCG logistics quality based on extended spanning tree.

On the number of leaves of a euclidean minimal spanning tree

1987 ◽  
Vol 24 (4) ◽  
pp. 809-826 ◽  
Author(s):  
J. Michael Steele ◽  
Lawrence A. Shepp ◽  
William F. Eddy
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Random Variables ◽  
Minimal Spanning Tree ◽  
Absolutely Continuous ◽  
Mean And Variance ◽  
Number Of Leaves ◽  
The Mean ◽  
Vertex Degrees ◽  
Minimal Spanning Trees

Let Vk,n be the number of vertices of degree k in the Euclidean minimal spanning tree of Xi, , where the Xi are independent, absolutely continuous random variables with values in Rd. It is proved that n–1Vk,n converges with probability 1 to a constant α k,d. Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of Vk,n.

scholarly journals Matchings, Cycle Bases, and the Maximum Genus of a Graph

10.37236/2479 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Michal Kotrbčík ◽  
Martin Škoviera
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Perfect Matching ◽  
Intersection Graph ◽  
Matching Number ◽  
Maximum Genus ◽  
Cycle Space ◽  
Intersection Graphs ◽  
Connected Graphs ◽  
Cycle Bases

We study the interplay between the maximum genus of a graph and bases of its cycle space via the corresponding intersection graph. Our main results show that the matching number of the intersection graph is independent of the basis precisely when the graph is upper-embeddable, and completely describe the range of matching numbers when the graph is not upper-embeddable. Particular attention is paid to cycle bases consisting of fundamental cycles with respect to a given spanning tree. For $4$-edge-connected graphs, the intersection graph with respect to any spanning tree (and, in fact, with respect to any basis) has either a perfect matching or a matching missing exactly one vertex. We show that if a graph is not $4$-edge-connected, different spanning trees may lead to intersection graphs with different matching numbers. We also show that there exist $2$-edge connected graphs for which the set of values of matching numbers of their intersection graphs contains arbitrarily large gaps.

scholarly journals A Minimum-Spanning-Tree-Inspired Algorithm for Channel Assignment in 802.11 Networks

2016 ◽  
Vol 62 (4) ◽  
pp. 379-388 ◽  
Author(s):  
Iwona Dolińska ◽  
Mariusz Jakubowski ◽  
Antoni Masiukiewicz ◽  
Grzegorz Rządkowski ◽  
Kamil Piórczyński
Keyword(s):  
Greedy Algorithm ◽  
Channel Assignment ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Minimum Spanning Trees ◽  
Undirected Graphs ◽  
Interference Level ◽  
802.11 Networks ◽  
Crowded Environment

Abstract Channel assignment in 2.4 GHz band of 802.11 standard is still important issue as a lot of 2.4 GHz devices are in use. This band offers only three non-overlapping channels, so in crowded environment users can suffer from high interference level. In this paper, a greedy algorithm inspired by the Prim’s algorithm for finding minimum spanning trees (MSTs) in undirected graphs is considered for channel assignment in this type of networks. The proposed solution tested for example network distributions achieves results close to the exhaustive approach and is, in many cases, several orders of magnitude faster.

On the number of leaves of a euclidean minimal spanning tree

1987 ◽  
Vol 24 (04) ◽  
pp. 809-826 ◽  
Author(s):  
J. Michael Steele ◽  
Lawrence A. Shepp ◽  
William F. Eddy
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Random Variables ◽  
Minimal Spanning Tree ◽  
Absolutely Continuous ◽  
Mean And Variance ◽  
Number Of Leaves ◽  
The Mean ◽  
Vertex Degrees ◽  
Minimal Spanning Trees

Let Vk,n be the number of vertices of degree k in the Euclidean minimal spanning tree of Xi , , where the Xi are independent, absolutely continuous random variables with values in Rd. It is proved that n –1 Vk,n converges with probability 1 to a constant α k,d. Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of Vk,n.

Emergence of mono-cluster flocking in the thermomechanical Cucker–Smale model under switching topologies

2020 ◽  
pp. 1-38
Author(s):  
Jiu-Gang Dong ◽  
Seung-Yeal Ha ◽  
Doheon Kim
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Dynamical Variable ◽  
Switching Network ◽  
Logarithmic Function ◽  
System Parameters ◽  
Network Topologies ◽  
Isothermal Case ◽  
Uniformly Bounded ◽  
Emergent Dynamics

We study the emergent dynamics of the thermomechanical Cucker–Smale (TCS) model with switching network topologies. The TCS model is a generalized CS model with extra internal dynamical variable called “temperature” in which isothermal case exactly coincides with the CS model for flocking. In previous studies, emergent dynamics of the TCS model has been mostly restricted to some static network topologies such as complete graph, connected graph with positive in and out degrees at each node, and digraphs with spanning trees. In this paper, we consider switching network topologies with a spanning tree in a sequence of time-blocks, and present two sufficient frameworks leading to the asymptotic mono-cluster flocking in terms of initial data and system parameters. In the first framework in which the sizes of time-blocks are uniformly bounded by some positive constant, we show that temperature and velocity diameters tend to zero exponentially fast, and spatial diameter is uniformly bounded. In the second framework, we admit a situation in which the sizes of time-blocks may grow mildly by a logarithmic function. In latter framework, our temperature and velocity diameters tend to zero at least algebraically slow.

scholarly journals Random minimal directed spanning trees and Dickman-type distributions

2004 ◽  
Vol 36 (03) ◽  
pp. 691-714 ◽  
Author(s):  
Mathew D. Penrose ◽  
Andrew R. Wade
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Dirichlet Distribution ◽  
Maximal Length ◽  
Directed Path ◽  
Limiting Distributions ◽  
Limit Distributions ◽  
Tree Construction ◽  
Directed Spanning Tree ◽  
Westerly Direction

In Bhatt and Roy's minimal directed spanning tree construction fornrandom points in the unit square, all edges must be in a south-westerly direction and there must be a directed path from each vertex to the root placed at the origin. We identify the limiting distributions (for largen) for the total length of rooted edges, and also for the maximal length of all edges in the tree. These limit distributions have been seen previously in analysis of the Poisson-Dirichlet distribution and elsewhere; they are expressed in terms of Dickman's function, and their properties are discussed in some detail.

scholarly journals On a random directed spanning tree

2004 ◽  
Vol 36 (1) ◽  
pp. 19-42 ◽  
Author(s):  
Abhay G. Bhatt ◽  
Rahul Roy
Keyword(s):  
Spanning Tree ◽  
Poisson Point Process ◽  
Spanning Trees ◽  
Asymptotic Properties ◽  
Record Values ◽  
Extreme Value Statistics ◽  
Minimal Spanning Tree ◽  
Unit Square ◽  
Directed Spanning Tree ◽  
Number Of Branches

We study the asymptotic properties of a minimal spanning tree formed by n points uniformly distributed in the unit square, where the minimality is amongst all rooted spanning trees with a direction of growth. We show that the number of branches from the root of this tree, the total length of these branches, and the length of the longest branch each converges weakly. This model is related to the study of record values in the theory of extreme-value statistics and this relation is used to obtain our results. The results also hold when the tree is formed from a Poisson point process of intensity n in the unit square.

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