Graph Theory

This book contains the successful invited submissions [1–10] to a special issue of Symmetry on the subject area of ‘graph theory’ [...

Residual effect of natural and synthetic zinc chelates on zinc in a soil solution of a waterlogged acidic soil. Evolution of the pH and redox potential.

Zinc chelates have been widely used to correct deficiencies in this micronutrient in different soil types and under different moisture conditions. The aging of the metal in soil could cause a change in its availability. Over time the most labile forms of Zn could decrease in activity and extractability and change to more stable forms. Various soil parameters, such as redox conditions, time, soil type and moisture conditions, affect the aging process and modify the solubility of the metal. In general, redox conditions influence pH and also the chemical forms dissolved in the soil solution. Soil pH also affects Zn solubility; at high pH values, most of the Zn is present in forms that are not bioavailable to plants. The objective of this study was to determine the changes in Zn over time in a soil solution in a waterlogged acidic soil to which synthetic and natural chelates were applie

Measurable diagonalization of positive definite matrices

In this paper we show that any positive definite matrix V with measurable entries can be written as V = U Lambda U*, where the matrix Lambda is diagonal, the matrix U is unitary, and the entries of U and Lambda are measurable functions (U* denotes the transpose conjugate of U). This result allows to obtain results about the zero location and asymptotic behavior of extremal polynomials with respect to a generalized non-diagonal Sobolev norm in which products of derivatives of different order appear. The orthogonal polynomials with respect to this Sobolev norm are a particular case of those extremal polynomials.The first author was partially supported by a grant from Ministerio de Economía y Competitividad. Dirección General de Investigación Científica y Técnica (MTM2012-36732-03-01), Spain. The second author was partially supported by a grant from CONACYT (CONACYT-UAGI0110/62/10 FON.INST.8/10), México

New results on the harmonic index and its generalizations

In this paper we obtain new inequalities involving the harmonic index and the(general) sum-connectivity index, and characterize graphs extremal with respect tothem. In particular, we improve and generalize some known inequalities and werelate this indices to other well-known topological indices.The authors are grateful to the referees for their valuable comments which have improved this paper. This work is supported in part by two grants from Ministerio de Economía y Competititvidad (MTM2013-46374-P and MTM2015-69323-REDT), Spain, and a grant from CONACYT (FOMIX-CONACyT-UAGro 249818), México

Spectral properties of geometric-arithmetic index

The concept of geometric-arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. One of the main aims of algebraic graph theory is to determine how, or whether, properties of graphs are reflected in the algebraic properties of some matrices. The aim of this paper is to study the geometric-arithmetic index GA(1) from an algebraic viewpoint. Since this index is related to the degree of the vertices of the graph, our main tool will be an appropriate matrix that is a modification of the classical adjacency matrix involving the degrees of the vertices. Moreover, using this matrix, we define a GA Laplacian matrix which determines the geometric-arithmetic index of a graph and satisfies properties similar to the ones of the classical Laplacian matrix. (C) 2015 Elsevier Inc. All rights reserved.This research was supported in part by a Grant from Ministerio de Economía y Competitividad (MTM 2013-46374-P), Spain, and a Grant from CONACYT (FOMIX-CONACyT-UAGro 249818), México

Spectral study of the Geometric-Arithmetic Index

The concept of geometric-arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. One of the main aims of algebraic graph theory is to determine how, or whether, properties of graphs are reected in the algebraic properties of some matrices. The aim of this paper is to study the geometric-arithmetic index GA1 from an algebraic viewpoint. Since this index is related to the degree of the vertices of the graph, our main tool will be an appropriate matrix that is a modification of the classical adjacency matrix involving the degrees of the vertices.Supported in part by a grant from Ministerio de Economía y Competititvidad (MTM 2013-46374-P), Spain, and by a grant from CONACYT (CONACYT-UAG I0110/62/10), México.Publicad

On the Geometric-Arithmetic Index

The concept of geometric-arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. The aim of this paper is to obtain new inequalities involving the geometric-arithmetic index GA1 and characterize graphs extremal with respect to them. In particular, we improve some known inequalities and we relate GA1 to other well known topological indices.Publicad

Optimal upper bounds of the geometric-arithmetic index

The concept of geometric-arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. The aim of this paper is to obtain new upper bounds of the geometric-arithmetic index and characterize graphs extremal with respect to them.This research was supported by a grant from Agencia Estatal de Investigación (PID2019-106433GB-I00 / AEI / 10.13039/501100011033), Spain

A note on isoperimetric inequalities of Gromov hyperbolic manifolds and graphs

We study in this paper the relationship of isoperimetric inequality and hyperbolicity for graphs and Riemannian manifolds. We obtain a characterization of graphs and Riemannian manifolds (with bounded local geometry) satisfying the (Cheeger) isoperimetric inequality, in terms of their Gromov boundary, improving similar results from a previous work. In particular, we prove that having a pole is a necessary condition to have isoperimetric inequality and, therefore, it can be removed as hypothesis.First author supported in part by a Grant from Ministerio de Ciencia, Innovación y Universidades (PGC2018-098321-B-I00), Spain. Second author supported in part by two Grants from Ministerio de Economía y Competitividad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2017-90584-REDT), Spain. Also, the research of the second author was supported by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation)

New Bounds for Topological Indices on Trees through Generalized Methods

This article belongs to the Special Issue Analytical and Computational Properties of Topological Indices.Topological indices are useful for predicting the physicochemical behavior of chemical compounds. A main problem in this topic is finding good bounds for the indices, usually when some parameters of the graph are known. The aim of this paper is to use a unified approach in order to obtain several new inequalities for a wide family of topological indices restricted to trees and to characterize the corresponding extremal trees. The main results give upper and lower bounds for a large class of topological indices on trees, fixing or not the maximum degree. This class includes the first variable Zagreb, the Narumi–Katayama, the modified Narumi–Katayama and the Wiener index.The first author was partially supported by a grant from Ministerio de Ciencia, Innovación y Universidades (PGC2018-098321-B-I00), Spain; the second author was partially supported by two grants from Ministerio de Economía y Competitividad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2017-90584-REDT), Spain