scholarly journals Wiener Complexity versus the Eccentric Complexity

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 79
Author(s):  
Martin Knor ◽  
Riste Škrekovski
Keyword(s):  
Unicyclic Graph ◽  
The Other ◽  
Unicyclic Graphs ◽  
Cyclomatic Number ◽  
Open Question ◽  
Sum Of Distances

Let wG(u) be the sum of distances from u to all the other vertices of G. The Wiener complexity, CW(G), is the number of different values of wG(u) in G, and the eccentric complexity, Cec(G), is the number of different eccentricities in G. In this paper, we prove that for every integer c there are infinitely many graphs G such that CW(G)−Cec(G)=c. Moreover, we prove this statement using graphs with the smallest possible cyclomatic number. That is, if c≥0 we prove this statement using trees, and if c<0 we prove it using unicyclic graphs. Further, we prove that Cec(G)≤2CW(G)−1 if G is a unicyclic graph. In our proofs we use that the function wG(u) is convex on paths consisting of bridges. This property also promptly implies the already known bound for trees Cec(G)≤CW(G). Finally, we answer in positive an open question by finding infinitely many graphs G with diameter 3 such that Cec(G)<CW(G).

Unicyclic graphs with regular endomorphism monoids

2016 ◽  
Vol 08 (02) ◽  
pp. 1650020 ◽  
Author(s):  
Xiaobin Ma ◽  
Dein Wong ◽  
Jinming Zhou
Keyword(s):  
Unicyclic Graph ◽  
Endomorphism Monoid ◽  
Unicyclic Graphs ◽  
Endomorphism Monoids ◽  
Odd Cycle ◽  
Open Question

The motivation of this paper comes from an open question: which graphs have regular endomorphism monoids? In this paper, we give a definitely answer for unicyclic graphs, proving that a unicyclic graph [Formula: see text] is End-regular if and only if, either [Formula: see text] is an even cycle with 4, 6 or 8 vertices, or [Formula: see text] contains an odd cycle [Formula: see text] such that the distance of any vertex to [Formula: see text] is at most 1, i.e., [Formula: see text]. The join of two unicyclic graphs with a regular endomorphism monoid is explicitly described.

scholarly journals The signless Laplacian coefficients and the incidence energy of graphs with a given bipartition

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 4215-4232
Author(s):  
Lei Zhong ◽  
Wen-Huan Wang
Keyword(s):  
Minimal Element ◽  
Unicyclic Graph ◽  
The Other ◽  
Unicyclic Graphs ◽  
Graph Transformations ◽  
Signless Laplacian ◽  
Positive Integers ◽  
Quasi Ordering

We consider two classes of the graphs with a given bipartition. One is trees and the other is unicyclic graphs. The signless Laplacian coefficients and the incidence energy are investigated for the sets of trees/unicyclic graphs with n vertices in which each tree/unicyclic graph has an (n1,n2)-bipartition, where n1 and n2 are positive integers not less than 2 and n1+n2 = n. Four new graph transformations are proposed for studying the signless Laplacian coefficients. Among the sets of trees/unicyclic graphs considered, we obtain exactly, for each, the minimal element with respect to the quasi-ordering according to their signless Laplacian coefficients and the element with the minimal incidence energies.

Incarnation as Awakening: Katsumi Takizawa Reading Karl Barth

Exchange ◽  
2007 ◽  
Vol 36 (2) ◽  
pp. 144-155
Author(s):  
Susanne Hennecke
Keyword(s):  
Karl Barth ◽  
The Other ◽  
Historical Jesus ◽  
Kyoto School ◽  
Dialectical Theology ◽  
Christian Theologian ◽  
The One ◽  
Open Question

AbstractThis contribution deals with the thinking of the Buddhist philosopher and Christian theologian Katsumi Takizawa (1909-1984) on incarnation. Firstly, it gives a short biographical and theological introduction to Takizawa, who was influenced not only by the "father" of the so-called dialectical theology, Karl Barth, but also by one of the famous figures of the Kyoto-school, the philosopher Kitaro Nishida.This contribution concentrates, secondly, on Takizawa's the-anthropological re-interpretation of the incarnation. It is argued that for Takizawa incarnation has to be seen as an awakening of the historical Jesus (or other historical phenomena) to what he calls the original fact: the eternal relationship between God and man.Thirdly, this contribution discusses the the-anthropological thinking of Takizawa about incarnation in five short points. Apart from the positive challenges of Takizawa's thinking especially for the theology of Karl Barth, it marks clearly the most thrilling point between Takizawa's thinking on the one side and that of scholars in Barthian theology on the other side. The open question that comes up is if incarnation really can be thought without a historical mediation or mediator, as Takizawa seems to claim.

Distance spectral radius of unicyclic graphs with fixed maximum degree

2019 ◽  
Vol 19 (04) ◽  
pp. 2050068
Author(s):  
Hezan Huang ◽  
Bo Zhou
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Maximum Degree ◽  
Distance Matrix ◽  
The Other ◽  
Connected Graphs ◽  
Largest Eigenvalue ◽  
Unicyclic Graphs ◽  
The Largest Eigenvalue

The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. For integers [Formula: see text] and [Formula: see text] with [Formula: see text], we prove that among the connected graphs on [Formula: see text] vertices of given maximum degree [Formula: see text] with at least one cycle, the graph [Formula: see text] uniquely maximizes the distance spectral radius, where [Formula: see text] is the graph obtained from the disjoint star on [Formula: see text] vertices and path on [Formula: see text] vertices by adding two edges, one connecting the star center with a path end, and the other being a chord of the star.

Framing Engineering Problems: Basic Concept

2014 ◽  
Author(s):  
Shuichi Fukuda
Keyword(s):  
Bounded Rationality ◽  
Basic Concept ◽  
The Other ◽  
Engineering Problems ◽  
Tremendous Amount ◽  
Open Question ◽  
Rational Knowledge

There are two kinds of knowledge in engineering. One is rational knowledge. We understand the phenomena and we can apply rational approaches. The other is knowledge about phenomena which we do not understand well, but which we can control and utilize for engineering. For example, we do not understand arc phenomena well, although there are tremendous amount of work about arc. If we do, we could prevent thunder and lightning. However, we utilize arc for welding. Without arc, most of our bridges, buildings, etc would not have been built. As Engineering is a DO activity and we do not have to UNDERSTAND the phenomena as scientists do. What is very important in engineering is how we can utilize such knowledge about these phenomena, which we do not understand well, but which we can control. And to develop a safe and secure system, we have to let them work in good harmony. This is the problem of frames as AI researchers call it. Although this is still an open question in AI, engineers have to go beyond the bounded rationality. This paper describes the basic concept of how we engineers could possibly tackle this problem.

scholarly journals When is a dynamical system mean sensitive?

2017 ◽  
Vol 39 (06) ◽  
pp. 1608-1636 ◽  
Author(s):  
FELIPE GARCÍA-RAMOS ◽  
JIE LI ◽  
RUIFENG ZHANG
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
The Other ◽  
Positive Entropy ◽  
Topological Dynamical System ◽  
Other Hand ◽  
Open Question ◽  
Uniquely Ergodic ◽  
Mean Sensitivity ◽  
Transitive System

This article is devoted to studying which conditions imply that a topological dynamical system is mean sensitive and which do not. Among other things, we show that every uniquely ergodic, mixing system with positive entropy is mean sensitive. On the other hand, we provide an example of a transitive system which is cofinitely sensitive or Devaney chaotic with positive entropy but fails to be mean sensitive. As applications of our theory and examples, we negatively answer an open question regarding equicontinuity/sensitivity dichotomies raised by Tu, we introduce and present results of locally mean equicontinuous systems and we show that mean sensitivity of the induced hyperspace does not imply that of the phase space.

scholarly journals Egalitarian men: stereotypes and discrimination in the labor market

2020 ◽  
Vol 23 (2) ◽  
pp. 111-147
Author(s):  
Hyalle Abreu Viana ◽  
Ana Raquel Rosas Torres ◽  
José Luis Álvaro Estriamana
Keyword(s):  
Labor Market ◽  
University Students ◽  
Gender Equality ◽  
The Other ◽  
Work Context ◽  
Family Responsibilities ◽  
Men And Women ◽  
Other Hand ◽  
Open Question ◽  
Do So

This article aimed to analyze the stereotypes attributed to "egalitarian men", understood here as men who support gender equality in relation to domestic and family responsibilities as well as inclusion in the workforce. To do so, two studies were carried out. The first study investigated the attribution of stereotypes to egalitarian men through a single open question. A total of 250 university students participated in this study, of which 51.1% were male, and their average age was 21.5 years (SD = 4.39). The second study analyzed the attribution of stereotypes to egalitarian or traditional men and women in a work context considered masculine. Participants included 221 university students with a mean age of 21.9 years (SD = 4.19), the majority (54.3%) being male. Taken together, the results of the two studies indicate that the egalitarian man is perceived as fragile and possibly homosexual. On the other hand, he is also seen as being more competent than traditional men.

scholarly journals On the Estrada Indices of Unicyclic Graphs with Fixed Diameters

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2395
Author(s):  
Wenjie Ning ◽  
Kun Wang
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Unicyclic Graph ◽  
Unicyclic Graphs ◽  
Estrada Index

The Estrada index of a graph G is defined as EE(G)=∑i=1neλi, where λ1,λ2,…,λn are the eigenvalues of the adjacency matrix of G. A unicyclic graph is a connected graph with a unique cycle. Let U(n,d) be the set of all unicyclic graphs with n vertices and diameter d. In this paper, we give some transformations which can be used to compare the Estrada indices of two graphs. Using these transformations, we determine the graphs with the maximum Estrada indices among U(n,d). We characterize two candidate graphs with the maximum Estrada index if d is odd and three candidate graphs with the maximum Estrada index if d is even.

scholarly journals Hamiltonian Cycles in Squares of Vertex-Unicyclic Graphs

1976 ◽  
Vol 19 (2) ◽  
pp. 169-172 ◽  
Author(s):  
Herbert Fleischner ◽  
Arthur M. Hobbs
Keyword(s):  
Sufficient Conditions ◽  
Unicyclic Graph ◽  
Necessary And Sufficient Conditions ◽  
Hamiltonian Cycles ◽  
Unicyclic Graphs ◽  
Necessary And Sufficient

In this paper we determine necessary and sufficient conditions for the square of a vertex-unicyclic graph to be Hamiltonian. The conditions are simple and easily checked. Further, we show that the square of a vertex-unicyclic graph is Hamiltonian if and only if it is vertex-pancyclic.

scholarly journals On the Wiener Complexity and the Wiener Index of Fullerene Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1071 ◽  
Author(s):  
Andrey A. Dobrynin ◽  
Andrei Yu Vesnin
Keyword(s):  
Mathematical Models ◽  
Wiener Index ◽  
The Other ◽  
Graph Invariant ◽  
Calculation Results ◽  
Local Graph ◽  
Sum Of Distances ◽  
Fullerene Graphs

Fullerenes are molecules that can be presented in the form of cage-like polyhedra, consisting only of carbon atoms. Fullerene graphs are mathematical models of fullerene molecules. The transmission of a vertex v of a graph is a local graph invariant defined as the sum of distances from v to all the other vertices. The number of different vertex transmissions is called the Wiener complexity of a graph. Some calculation results on the Wiener complexity and the Wiener index of fullerene graphs of order n ≤ 232 and IPR fullerene graphs of order n ≤ 270 are presented. The structure of graphs with the maximal Wiener complexity or the maximal Wiener index is discussed, and formulas for the Wiener index of several families of graphs are obtained.

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