Sharp bounds for the modified multiplicative zagreb indices of graphs with vertex connectivity at most k

© 2019, University of Nis. All rights reserved. Zagreb indices and their modified versions of a molecular graph originate from many practical problems such as two dimensional quantitative structure-activity (2D QSAR) and molecular chirality. Nowadays, they have become important invariants which can be used to characterize the properties of graphs from different aspects. Let Vkn (or Ekn respectively) be a set of graphs of n vertices with vertex connectivity (or edge connectivity respectively) at most k. In this paper, we explore some properties of the modified first and second multiplicative Zagreb indices of graphs in Vkn and Ekn. By using analytic and combinatorial tools, we obtain some sharp lower and upper bounds for these indices of graphs in Vkn and Ekn. In addition, the corresponding extremal graphs which attain the lower or upper bounds are characterized. Our results enrich outcomes on studying Zagreb indices and the methods developed in this paper may provide some new tools for investigating the values on modified multiplicative Zagreb indices of other classes of graphs

The First Zagreb Index, Vertex-Connectivity, Minimum Degree And Independent Number in Graphs

Let G be a simple, undirected and connected graph. Defined by M1(G) and RMTI(G) the first Zagreb index and the reciprocal Schultz molecular topological index of G, respectively. In this paper, we determined the graphs with maximal M1 among all graphs having prescribed vertex-connectivity and minimum degree, vertex-connectivity and bipartition, vertex-connectivity and vertex-independent number, respectively. As applications, all maximal elements with respect to RMTI are also determined among the above mentioned graph families, respectively

Symmetry and Complexity

Symmetry and complexity are the focus of a selection of outstanding papers, ranging from pure Mathematics and Physics to Computer Science and Engineering applications. This collection is based around fundamental problems arising from different fields, but all of them have the same task, i.e. breaking the complexity by the symmetry. In particular, in this Issue, there is an interesting paper dealing with circular multilevel systems in the frequency domain, where the analysis in the frequency domain gives a simple view of the system. Searching for symmetry in fractional oscillators or the analysis of symmetrical nanotubes are also some important contributions to this Special Issue. More papers, dealing with intelligent prognostics of degradation trajectories for rotating machinery in engineering applications or the analysis of Laplacian spectra for categorical product networks, show how this subject is interdisciplinary, i.e. ranging from theory to applications. In particular, the papers by Lee, based on the dynamics of trapped solitary waves for special differential equations, demonstrate how theory can help us to handle a practical problem. In this collection of papers, although encompassing various different fields, particular attention has been paid to the common task wherein the complexity is being broken by the search for symmetry

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view

Structure, entropy and evolution of systems of cities

One of the main questions in Urban Science is whether systems of cities around the world show similarities in their structure and trajectories of development. Shannon entropy has played a crucial role in this line of research, both because it is a versatile measure of uniformity and because of its ability to discriminate significant patterns from only seemingly organised maximum randomness. In this thesis, we present novel ways to analyse the structure of systems of cities and its evolution using entropy-based measures. We focus on key morphological aspects of a system of cities: the distribution of city sizes, their spatial arrangement, the population density and land use of their surroundings, and their connectivity via transport infrastructure; which we reconnect to human activities via spatial interaction models. We propose normalisation formulae for the first degree-based graph entropy that facilitate its interpretation as a measure of balance of the degree sequence of a network. We define a local entropy measure for raster data that quantifies the heterogeneity of a variable of interest in the surroundings of each cell. We define a measure of morphological polycentricity for historical systems of cities based on the entropy of the most likely potential interactions between the cities. We apply our methods to analyse systems of cities in different parts of the world and moments in history. We study the evolution of the entropy of city sizes in the main European powers from 1300 to 1850; the local entropy of land use and population density in Italy, the British Isles, and South Asia from 1700 to modern day; and the spatial organisation and morphological polycentricity of English and Welsh towns in the 19th century, via the entropy of spatial networks informed by the emerging railway system. Finally, we model the spatial-temporal dynamics of geo-tagged Tweets in London, of the Hungarian social network iWiW, and of the network of literary imitations between medieval Occitan troubadours

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

Advances in Stereo Vision

Stereopsis is a vision process whose geometrical foundation has been known for a long time, ever since the experiments by Wheatstone, in the 19th century. Nevertheless, its inner workings in biological organisms, as well as its emulation by computer systems, have proven elusive, and stereo vision remains a very active and challenging area of research nowadays. In this volume we have attempted to present a limited but relevant sample of the work being carried out in stereo vision, covering significant aspects both from the applied and from the theoretical standpoints