scholarly journals An application of graph theory to linguistic complexity

2014 ◽  
Vol 1 (1) ◽  
pp. 89-102
Author(s):  
Alexander Piperski
Keyword(s):  
Graph Theory ◽  
Noun Phrases ◽  
Complexity Measure ◽  
Vertex Degree ◽  
Linguistic Complexity ◽  
Dual Nature ◽  
Linguistic Sign ◽  
Average Vertex ◽  
Three Components

Abstract This article introduces a new measure of linguistic complexity which is based on the dual nature of the linguistic sign. Complexity is analyzed as consisting of three components, namely the conceptual complexity (complexity of the signified), the formal complexity (complexity of the signifier) and the form-meaning correspondence complexity. I describe a way of plotting the form-meaning relationship on a graph with two tiers (the form tier and the meaning tier) and apply a complexity measure from graph theory (average vertex degree) to assess the complexity of such graphs. The proposed method is illustrated by estimating the complexity of full noun phrases (determiner + adjective + noun) in English, Swedish, and German. I also mention the limitations and the problems which might arise when using this method.

scholarly journals On the density of Cayley graphs of R.Thompson’s group F in symmetric generators

2021 ◽  
pp. 1-13
Author(s):  
V. S. Guba
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Finite Graph ◽  
Vertex Degree ◽  
Doubling Property ◽  
Generating Set ◽  
Finite Set ◽  
Standard Set ◽  
Thompson’S Group ◽  
Average Vertex

By the density of a finite graph we mean its average vertex degree. For an [Formula: see text]-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with [Formula: see text] generators is amenable if and only if the density of the corresponding Cayley graph equals [Formula: see text]. A famous problem on the amenability of R. Thompson’s group [Formula: see text] is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators [Formula: see text], is at least [Formula: see text]. This estimate has not been exceeded so far. For the set of symmetric generators [Formula: see text], where [Formula: see text], the same example only gave an estimate of [Formula: see text]. There was a conjecture that for this generating set equality holds. If so, [Formula: see text] would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set [Formula: see text], the inequality [Formula: see text] holds. In this paper, we disprove this conjecture showing that the density of the Cayley graph of [Formula: see text] in symmetric generators [Formula: see text] strictly exceeds [Formula: see text]. Moreover, we show that even larger generating set [Formula: see text] does not have doubling property.

scholarly journals ON THE PROPERTIES OF THE CAYLEY GRAPH OF RICHARD THOMPSON'S GROUP F

2004 ◽  
Vol 14 (05n06) ◽  
pp. 677-702 ◽  
Author(s):  
V. S. GUBA
Keyword(s):  
Growth Rate ◽  
Lower Bound ◽  
Cayley Graph ◽  
Upper Bound ◽  
Growth Function ◽  
Vertex Degree ◽  
General Growth ◽  
Thompson’S Group ◽  
Average Vertex ◽  
Thompson's Group F

We study some properties of the Cayley graph of R. Thompson's group F in generators x0, x1. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3. It is known that a 2-generated group is not amenable if and only if the density of the corresponding Cayley graph is strictly less than 4. It is well known this is also equivalent to the existence of a doubling function on the Cayley graph. This means there exists a mapping from the set of vertices into itself such that for some constant K>0, each vertex moves by a distance at most K and each vertex has at least two preimages. We show that the density of the Cayley graph of a 2-generated group does not exceed 3 if and only if the group satisfies the above condition with K=1. Besides, we give a very easy formula to find the length (norm) of a given element of F in generators x0, x1. This simplifies the algorithm by Fordham. The length formula may be useful for finding the general growth function of F in generators x0, x1 and the growth rate of this function. In this paper, we show that the growth rate of F has a lower bound of [Formula: see text].

scholarly journals Limiting eigenvalue distribution of random matrices of Ihara zeta function of long-range percolation graphs

2018 ◽  
Vol 07 (03) ◽  
pp. 1850007
Author(s):  
O. Khorunzhiy
Keyword(s):  
Graph Theory ◽  
Zeta Function ◽  
Long Range ◽  
Riemann Hypothesis ◽  
Counting Function ◽  
Symmetric Matrices ◽  
Ihara Zeta Function ◽  
Unique Measure ◽  
Edge Probability ◽  
Average Vertex

We consider the ensemble of [Formula: see text] real random symmetric matrices [Formula: see text] obtained from the determinant form of the Ihara zeta function associated to random graphs [Formula: see text] of the long-range percolation radius model with the edge probability determined by a function [Formula: see text]. We show that the normalized eigenvalue counting function of [Formula: see text] weakly converges in average as [Formula: see text], [Formula: see text] to a unique measure that depends on the limiting average vertex degree of [Formula: see text] given by [Formula: see text]. This measure converges in the limit of infinite [Formula: see text] to a shift of the Wigner semi-circle distribution. We discuss relations of these results with the properties of the Ihara zeta function and weak versions of the graph theory Riemann Hypothesis.

Garbage Collection, Sunday Strolls, and Soldering Problems

1972 ◽  
Vol 65 (4) ◽  
pp. 307-309
Author(s):  
Walter Meyer
Keyword(s):  
High School ◽  
Graph Theory ◽  
Garbage Collection ◽  
Applied Mathematics ◽  
School Curriculum ◽  
High School Curriculum ◽  
Dual Nature ◽  
Practical Applications ◽  
Basic Ideas

Is there an area of mathematics that deals with garbage collection, Sunday strolls, and soldering problems all at once? Indeed there is, and it is called graph theory, a subject that considers the properties of configurations consisting of points and connecting lines such as the configuration shown in figure 2. (There is another meaning for the word graph, as in bar graph or graph of a function, which is not meant here.) The practical applications of graph theory are so widespread that this theory has become one of the most important and rapidly growing areas of applied mathematics in recent years. What is especially unique about it, however, is the extreme simplicity of the basic ideas. Because of this dual nature of practicality and simplicity, graphs have been creeping into the high school curriculum lately, often in the form of optional topics.

Creating recreational Hamiltonian cycle problems

2004 ◽  
Vol 88 (512) ◽  
pp. 215-218 ◽  
Author(s):  
Mark A. M. Lynch
Keyword(s):  
Hamiltonian Cycle ◽  
Vertex Degree ◽  
Hamiltonian Cycles ◽  
Arbitrary Size ◽  
Method Of Solution ◽  
Average Vertex

In this paper graphs that contain unique Hamiltonian cycles are introduced. The graphs are of arbitrary size and dense in the sense that their average vertex degree is greater than half the number of vertices that make up the graph. The graphs can be used to generate challenging puzzles. The problem is particularly challenging when the graph is large and the ‘method’ of solution is unknown to the solver.

scholarly journals Some Vertex/Edge-Degree-Based Topological Indices of r -Apex Trees

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Akbar Ali ◽  
Waqas Iqbal ◽  
Zahid Raza ◽  
Ekram E. Ali ◽  
Jia-Bao Liu ◽  
...  
Keyword(s):  
Graph Theory ◽  
Connected Graph ◽  
Nonnegative Integer ◽  
Symmetric Function ◽  
Topological Indices ◽  
Vertex Degree ◽  
Graph Invariants ◽  
Arbitrary Vertex ◽  
Chemical Graph ◽  
Chemical Graph Theory

In chemical graph theory, graph invariants are usually referred to as topological indices. For a graph G , its vertex-degree-based topological indices of the form BID G = ∑ u v ∈ E G β d u , d v are known as bond incident degree indices, where E G is the edge set of G , d w denotes degree of an arbitrary vertex w of G , and β is a real-valued-symmetric function. Those BID indices for which β can be rewritten as a function of d u + d v − 2 (that is degree of the edge u v ) are known as edge-degree-based BID indices. A connected graph G is said to be r -apex tree if r is the smallest nonnegative integer for which there is a subset R of V G such that R = r and G − R is a tree. In this paper, we address the problem of determining graphs attaining the maximum or minimum value of an arbitrary BID index from the class of all r -apex trees of order n , where r and n are fixed integers satisfying the inequalities n − r ≥ 2 and r ≥ 1 .

scholarly journals Revan and hyper-Revan indices of Octahedral and icosahedral networks

2018 ◽  
Vol 3 (1) ◽  
pp. 33-40 ◽  
Author(s):  
Abdul Qudair Baig ◽  
Muhammad Naeem ◽  
Wei Gao
Keyword(s):  
Graph Theory ◽  
Connected Graph ◽  
Vertex Degree ◽  
Chemical Graph ◽  
Chemical Graph Theory ◽  
Vertex Set

AbstractLet G be a connected graph with vertex set V(G) and edge set E(G). Recently, the Revan vertex degree concept is defined in Chemical Graph Theory. The first and second Revan indices of G are defined as R1(G) = $\begin{array}{} \displaystyle \sum\limits_{uv\in E} \end{array}$[rG(u) + rG(v)] and R2(G) = $\begin{array}{} \displaystyle \sum\limits_{uv\in E} \end{array}$[rG(u)rG(v)], where uv means that the vertex u and edge v are adjacent in G. The first and second hyper-Revan indices of G are defined as HR1(G) = $\begin{array}{} \displaystyle \sum\limits_{uv\in E} \end{array}$[rG(u) + rG(v)]2 and HR2(G) = $\begin{array}{} \displaystyle \sum\limits_{uv\in E} \end{array}$[rG(u)rG(v)]2. In this paper, we compute the first and second kind of Revan and hyper-Revan indices for the octahedral and icosahedral networks.

-BANHATTI, HYPER -BANHATTI INDICES AND THEIR POLYNOMIALS OF CERTAIN NETWORKS

2021 ◽  
Vol 10 (3) ◽  
pp. 1-8
Keyword(s):  
Graph Theory ◽  
Molecular Graph ◽  
Vertex Degree ◽  
Chemical Graph ◽  
Chemical Graph Theory

Recently, the -vertex degree concept was defined in Chemical Graph Theory. In this paper, we propose the first and second -Banhatti indices, first and second hyper -Banhatti indices and their corresponding polynomials of a molecular graph and compute exact formulas for silicate networks and hexagonal networks.

The Jovian ring

1984 ◽  
Vol 75 ◽  
pp. 203-209
Author(s):  
Joseph A. Burns
Keyword(s):  
Equatorial Plane ◽  
Charged Particles ◽  
Dynamical Evolution ◽  
Main Band ◽  
Small Grains ◽  
Three Components

ABSTRACTLying in Jupiter's equatorial plane is a diaphanous ring having little substructure within its three components (main band, faint disk, and halo). Micron-sized grains account for much of the visible ring, but particles of centimeter sizes and larger must also be present to absorb charged particles. Since dynamical evolution times and survival life times are quite short (≲102-3yr) for small grains, the Jovian ring is being continually replenished; probably most of the visible ring is generated by micrometeoroids colliding into unseen parent bodies that reside in the main band.

The dual nature of obesity in metabolic programming: quantity versus quality of adipose tissue

2020 ◽  
Vol 134 (18) ◽  
pp. 2447-2451
Author(s):  
Anissa Viveiros ◽  
Gavin Y. Oudit
Keyword(s):  
Adipose Tissue ◽  
Environmental Factors ◽  
Maternal Obesity ◽  
Metabolic Programming ◽  
Dual Nature ◽  
Prevalence Of Obesity ◽  
Adipose Tissue Volume ◽  
Males And Females ◽  
Metabolically Healthy

Abstract The global prevalence of obesity has been rising at an alarming rate, accompanied by an increase in both childhood and maternal obesity. The concept of metabolic programming is highly topical, and in this context, describes a predisposition of offspring of obese mothers to the development of obesity independent of environmental factors. Research published in this issue of Clinical Science conducted by Litzenburger and colleagues (Clin. Sci. (Lond.) (2020) 134, 921–939) have identified sex-dependent differences in metabolic programming and identify putative signaling pathways involved in the differential phenotype of adipose tissue between males and females. Delineating the distinction between metabolically healthy and unhealthy obesity is a topic of emerging interest, and the precise nature of adipocytes are key to pathogenesis, independent of adipose tissue volume.

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