scholarly journals The G-Convexity and the G-Centroids of Composite Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1927
Author(s):  
Prakash Veeraraghavan

The graph centroids defined through a topological property of a graph called g-convexity found its application in various fields. They have classified under the “facility location” problem. However, the g-centroid location for an arbitrary graph is NP-hard. Thus, it is necessary to devise an approximation algorithm for general graphs and polynomial-time algorithms for some special classes of graphs. In this paper, we study the relationship between the g-centroids of composite graphs and their factors under various well-known graph operations such as graph Joins, Cartesian products, Prism, and the Corona. For the join of two graphs G1 and G2, the weight sequence of the composite graph does not depend on the weight sequences of its factors; rather it depends on the incident pattern of the maximum cliques of G1 and G2. We also characterize the structure of the g-centroid under various cases. For the Cartesian product of G1 and G2 and the prism of a graph, we establish the relationship between the g-centroid of a composite graph and its factors. Our results will facilitate the academic community to focus on the factor graphs while designing an approximate algorithm for a composite graph.


2019 ◽  
Vol 17 (1) ◽  
pp. 1269-1280 ◽  
Author(s):  
Csilla Bujtás ◽  
Pakanun Dokyeesun ◽  
Vesna Iršič ◽  
Sandi Klavžar

Abstract The connected domination game on a graph G is played by Dominator and Staller according to the rules of the standard domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of G. If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the connected game domination number of G. Here this invariant is studied on Cartesian product graphs. A general upper bound is proved and demonstrated to be sharp on Cartesian products of stars with paths or cycles. The connected game domination number is determined for Cartesian products of P3 with arbitrary paths or cycles, as well as for Cartesian products of an arbitrary graph with Kk for the cases when k is relatively large. A monotonicity theorem is proved for products with one complete factor. A sharp general lower bound on the connected game domination number of Cartesian products is also established.



2004 ◽  
Vol 14 (01) ◽  
pp. 61-73 ◽  
Author(s):  
ROBERT ELSÄSSER ◽  
BURKHARD MONIEN ◽  
ROBERT PREIS ◽  
ANDREAS FROMMER

We discuss nearest neighbor load balancing schemes on processor networks which are represented by a cartesian product of graphs and present a new optimal diffusion scheme for general graphs. In the first part of the paper, we introduce the Alternating-Direction load balancing scheme, which reduces the number of load balance iterations by a factor of 2 for cartesian products of graphs. The resulting flow is theoretically analyzed and can be very high for certain cases. Therefore, we further present the Mixed-Direction scheme which needs the same number of iterations but computes in most cases a much smaller flow. In the second part of the paper, we present a simple optimal diffusion scheme for general graphs, calculating a balancing flow which is minimal in the l2 norm. It is based on the spectra of the graph representing the network and needs only m-1 iterations to balance the load with m being the number of distinct eigenvalues. Known optimal diffusion schemes have the same performance, however the optimal scheme presented in this paper can be implemented in a very simple manner. The number of iterations of optimal diffusion schemes is independent of the load scenario and, thus, they are practical for networks which represent graphs with known spectra. Finally, our experiments exhibit that the new optimal scheme can successfully be combined with the Alternating-Direction and Mixed-Direction schemes for efficient load balancing on product graphs.



10.37236/141 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Feng-Gen Lin ◽  
Lian-Zhu Zhang

The importance of Pfaffian orientations stems from the fact that if a graph $G$ is Pfaffian, then the number of perfect matchings of $G$ (as well as other related problems) can be computed in polynomial time. Although there are many equivalent conditions for the existence of a Pfaffian orientation of a graph, this property is not well-characterized. The problem is that no polynomial algorithm is known for checking whether or not a given orientation of a graph is Pfaffian. Similarly, we do not know whether this property of an undirected graph that it has a Pfaffian orientation is in NP. It is well known that the enumeration problem of perfect matchings for general graphs is NP-hard. L. Lovász pointed out that it makes sense not only to seek good upper and lower bounds of the number of perfect matchings for general graphs, but also to seek special classes for which the problem can be solved exactly. For a simple graph $G$ and a cycle $C_n$ with $n$ vertices (or a path $P_n$ with $n$ vertices), we define $C_n$ (or $P_n)\times G$ as the Cartesian product of graphs $C_n$ (or $P_n$) and $G$. In the present paper, we construct Pfaffian orientations of graphs $C_4\times G$, $P_4\times G$ and $P_3\times G$, where $G$ is a non bipartite graph with a unique cycle, and obtain the explicit formulas in terms of eigenvalues of the skew adjacency matrix of $\overrightarrow{G}$ to enumerate their perfect matchings by Pfaffian approach, where $\overrightarrow{G}$ is an arbitrary orientation of $G$.



2021 ◽  
Vol 37 (3) ◽  
pp. 907-917
Author(s):  
Martin Kreh ◽  
Jan-Hendrik de Wiljes

AbstractIn 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. In the same article, the authors proved some results on the solvability of Cartesian products, given solvable or distance 2-solvable graphs. We extend these results to Cartesian products of certain unsolvable graphs. In particular, we prove that ladders and grid graphs are solvable and, further, even the Cartesian product of two stars, which in a sense are the “most” unsolvable graphs.



2021 ◽  
Vol 2 ◽  
pp. 144-172
Author(s):  
Margarita Mikhailovna Pavlova ◽  

This publication introduces the academic community to the new data from the archives of the Merezhkovskys and D. V. Filosofov, which are stored in the Amherst Center for Russian Culture (Massachusetts, USA). It presents a selection of letters by Filosofov (1906) and N. A. Berdiaev (1907) that provide new details concerning the relationship between the two. Excerpts from the «diaries» of T. N. Gippius that supplement the «Berdiaev» narrative are published as an appendix.



Author(s):  
Mingyou Wu ◽  
Xi Li ◽  
Zhihao Liu ◽  
Hanwu Chen

The continuous-time quantum walk (CTQW) provides a new approach to problems in graph theory. In this paper, the correlation between the CTQW and cliques in graphs is studied, and an approximate algorithm for the maximum clique problem (MCP) based on the CTQW is given. Via both numerical and theoretical analyses, it is found that the maximum clique is related to the transmission characteristics of the CTQW on some special graphs. For general graphs, the correlation is difficult to describe analytically. Therefore, the transmission characteristics of the CTQW are applied as a vertex selection criterion to a classical MCP algorithm and it is compared with the original algorithm. Numerous simulation on general graphs shows that the new algorithm is more efficient. Furthermore, an approximate MCP algorithm based on the CTQW is introduced, which only requires a very small number of searches with a high approximation ratio.



Author(s):  
Dake Jiang ◽  
Hong Zhang ◽  
Lei Wang ◽  
Jin Liu ◽  
Linlin Chang

In the business ecosystem, the core business niche is not occupied by a structural position; its objectives and strategies are easily dispersed, while the core business can't coordinate the business ecosystem elements. Therefore, the ecological advantages obtained by core enterprises in the process of self-organization evolution are not sustainable. To solve this problem, a possible way is to explore how core enterprises consolidate and form new ecological advantages from the perspective of cross-border business. However, the academic community has not discussed the evolution path of cross-border entrepreneurship in detail. For this reason, based on the perspective of ecological advantages, this chapter discusses the strategic path of cross-border entrepreneurship of core enterprises and constructs an interaction model between ecological advantages and core enterprises' cross-border entrepreneurial paths. The study broadens the understanding of the relationship between corporate strategies and business ecosystems, then provides theoretical value for subsequent research.



1976 ◽  
Vol 22 (2) ◽  
pp. 212-220 ◽  
Author(s):  
D. A. Holton ◽  
J. A. Sims

AbstractWe consider the effect on the stability properties of a graph G, of the presence in the automorphism group of G of automorphisms (uv)h, where u and v are vertices of G, and h is a permutation of vertices of G excluding u and v. We find sufficient conditions for an arbitrary graph and a cartesian product to have stability index one, and conjecture in the latter case that they are necessary. Finally we exhibit explicitly a large class of graphs which have stability index one.



Author(s):  
Lina Morgado ◽  
Angelina Costa

Resumo: O ensino superior vive complexos momentos de mudança à escala mundial. O desenvolvimento exponencial das tecnologias e, consequentemente, da educação a distância e do elearning, os novos públicos que têm acesso a este nível de ensino, o facto de ser, pela sua natureza, um espaço de inovação na relação e construção do conhecimento, os desenvolvimentos científicos, concretamente na área da educação e das tecnologias, são fatores que têm feito emergir necessidades de análise, de avaliação e de reflexão da produção científica, tornando-as tarefas substanciais da comunidade académica. Com este artigo pretende-se esboçar alguns cenários de futuro na educação a distância e elearning, no ensino superior, olhando para a situação de Portugal no contexto europeu. Foram definidos três momentos de investigação. Num primeiro momento, pretende-se delinear o estado da arte da investigação sobre educação a distância e elearning, no ensino superior, mas ir para além dele, num segundo momento, convocando a discussão e a reflexão dos protagonistas no processo. Num terceiro momento, pretende-se confrontar as perspetivas nacionais com as perspetivas de especialistas europeus neste domínio. Os resultados que agora se apresentam constituem ainda um trabalho em progresso e por isso com resultados preliminares e parciais da primeira fase de investigação.Palavras-chave: educação a distância, elearning, ensino superior, investigação, gestão, inovação Abstract: Higher education experiments complex moments of change to a worldwide extent. With the exponential development of technology and, therefore, the distance education and learning, where a new public has access to this level of education. Other factors like being, by its nature, a space for innovation in the relationship and construction of knowledge, scientific developments, particularly in education and technology, contribute to emerging needs analysis, evaluation and reflection of the scientific production, making it the substantial tasks of the academic community. This study aims to outline scenarios of future in distance education and e-learning, in higher education, looking at Portugal's position in the European context. It has been defined three stages of the investigation. Initially, it is intended to outline the state of the art of research on distance education and e-learning, in higher education, but go beyond, in a second moment, summoning discussion and reflection of the protagonists in the process. Thirdly, the prospects of national and European perspectives of specialists in this field will be faced up. Still, in the research process, we present here the global design of the study and preliminary and partial results of phase one of the research.Keywords: distance education, e-learning, higher education, research projects, management, innovation Resumen: La enseñanza superior vive complejos momentos de cambio a escala mundial. El desarrollo exponencial de las tecnologías y, por consiguiente, de la educación a distancia y del elearning, los nuevos públicos que tienen acceso a este nivel de enseñanza, el hecho de ser, por su naturaleza, un espacio de innovación en la relación y la construcción del conocimiento, los desarrollos científicos, concretamente en el ámbito de la educación y de las tecnologías, son factores que han hecho emerger necesidades de análisis, de evaluación y de reflexión de la producción científica, haciéndolas tareas sustanciales de la comunidad académica. Con este artículo se pretende esbozar algunos escenarios de futuro en la educación a distancia y e-learning en la educación superior, mirando la situación de Portugal en el contexto europeo. Se definieron tres momentos de investigación. En un primer momento, se pretende delinear el estado del arte de la investigación sobre educación a distancia y elearning, en la enseñanza superior, pero ir más allá de él, en un segundo momento, convocando la discusión y la reflexión de los protagonistas en el proceso. En un tercer momento, se pretende confrontar las perspectivas nacionales con las perspectivas de expertos europeos en este ámbito. Los resultados que ahora se presentan constituyen todavía un trabajo en progreso y por ello con resultados preliminares y parciales de la primera fase de investigación.Palabras-clave: educación a distancia, elearning, enseñanza superior, investigación, gestión, innovación



Author(s):  
Md. Mizanur Rahman ◽  
Noor Azman Ali ◽  
Amer Hamzah Jantan ◽  
Zuraina Dato' Mansor ◽  
Md. Saidur Rahaman

Purpose The purpose of this study is to confirm that work to family conflict (W to FC), family to work conflict (F to WC) and work family balance (WFB) are the predictors of job satisfaction (JS) for Malaysian academic community. To achieve the fundamental objective, the authors first test the direct (structural model) effects of W to FC and F to WC on JS. Secondly, using mediation model, the indirect effects of WFB were calculated through W to FC, F to WC and JS. Design/methodology/approach Using convenience sampling, the data was collected from 280 academic people who work at private universities in Malaysia. Structural equation modelling technique was applied to complete the data analyses procedures. Findings The findings revealed that W to FC and F to WC have negative significant effects on JS. Besides, WFB partially mediates only the relationship between W to FC and JS while no mediation effect was found for F to WC and JS. Research limitations/implications All the research variables in this study were individual-level variables, thus it is suggested to use some organizational and national level variables such as religion and culture as they might be good intervening variables for predicting JS. Practical implications Using the findings of this research, the Malaysian private universities community can take some necessary initiatives to mitigate work family conflict (W-FC) and ensure WFB and JS that might enhance the standard of higher education in Malaysia. Originality/value To the best of the authors’ knowledge, this is the first attempt to test the mediation effect of WFB in the relationship between both directions of W-FC and JS of employees from the perspective of Malaysia (a collectivist community).



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