1,791 research outputs found
New inequalities involving the geometric-arithmetic index
Let G = (V, E) be a simple connected graph and di be the degree of its ith
vertex. In a recent paper [J. Math. Chem. 46 (2009) 1369-1376] the first geometricarithmetic index of a graph G was defined as
GA1 = X
uvâE
2
â
dudv
du + dv
.
This graph invariant is useful for chemical proposes. The main use of GA1 is for designing so-called quantitative structure-activity relations and quantitative structureproperty relations. In this paper we obtain new inequalities involving the geometricarithmetic index GA1 and characterize the graphs which make the inequalities tight.
In particular, we improve some known results, generalize other, and we relate GA1
to other well-known topological indices.We are grateful to the constructive comments from anonymous referee on our pape. The first and third authors are supported by the "Ministerio de EconomĂa y Competititvidad" (MTM2013-46374-P and MTM2015-69323-REDT), Spain, and by the CONACYT (FOMIX-CONACyT-UAGro 249818), Mexico
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 EconomĂa y Competitividad (MTM 2013-46374-P), Spain, and a Grant from CONACYT (FOMIX-CONACyT-UAGro 249818), MĂ©xico
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
f-polynomial on some graph operations
Given any function f:Z+âR+ , let us define the f-index If(G)=âuâV(G)f(du) and the f-polynomial Pf(G,x)=âuâV(G)x1/f(du)â1, for x>0 . In addition, we define Pf(G,0)=limxâ0+Pf(G,x) . We use the f-polynomial of a large family of topological indices in order to study mathematical relations of the inverse degree, the generalized first Zagreb, and the sum lordeg indices, among others. In this paper, using this f-polynomial, we obtain several properties of these indices of some classical graph operations that include corona product and join, line, and Mycielskian, among others.Supported in part by two grants from the Ministerio de EconomĂa y
Competititvidad, Agencia Estatal de InvestigaciĂłn (AEI), and Fondo Europeo de Desarrollo Regional (FEDER)
(MTM2016-78227-C2-1-P and MTM2017-90584-REDT), Spain
Predicting Proteome-Early Drug Induced Cardiac Toxicity Relationships (Pro-EDICToRs) with Node Overlapping Parameters (NOPs) of a new class of Blood Mass-Spectra graphs
The 11th International Electronic Conference on Synthetic Organic Chemistry session Computational ChemistryBlood Serum Proteome-Mass Spectra (SP-MS) may allow detecting Proteome-Early Drug Induced Cardiac Toxicity Relationships (called here Pro-EDICToRs). However, due to the thousands of proteins in the SP identifying general Pro-EDICToRs patterns instead of a single protein marker may represents a more realistic alternative. In this sense, first we introduced a novel Cartesian 2D spectrum graph for SP-MS. Next, we introduced the graph node-overlapping parameters (nopk) to numerically characterize SP-MS using them as inputs to seek a Quantitative Proteome-Toxicity Relationship (QPTR) classifier for Pro-EDICToRs with accuracy higher than 80%. Principal Component Analysis (PCA) on the nopk values present in the QPTR model explains with one factor (F1) the 82.7% of variance. Next, these nopk values were used to construct by the first time a Pro-EDICToRs Complex Network having nodes (samples) linked by edges (similarity between two samples). We compared the topology of two sub-networks (cardiac toxicity and control samples); finding extreme relative differences for the re-linking (P) and Zagreb (M2) indices (9.5 and 54.2 % respectively) out of 11 parameters. We also compared subnetworks with well known ideal random networks including Barabasi-Albert, Kleinberg Small World, Erdos-Renyi, and Epsstein Power Law models. Finally, we proposed Partial Order (PO) schemes of the 115 samples based on LDA-probabilities, F1-scores and/or network node degrees. PCA-CN and LDA-PCA based POs with Tanimotoâs coefficients equal or higher than 0.75 are promising for the study of Pro-EDICToRs. These results shows that simple QPTRs models based on MS graph numerical parameters are an interesting tool for proteome researchThe authors thank projects funded by the Xunta de Galicia (PXIB20304PR and BTF20302PR) and the Ministerio de Sanidad y Consumo (PI061457). GonzĂĄlez-DĂaz H. acknowledges tenure track research position funded by the Program Isidro Parga Pondal, Xunta de Galici
On K-trees and Special Classes of K-trees
The class of k-trees is defined recursively as follows: the smallest k-tree is the k-clique. If G is a graph obtained by attaching a vertex v to a k-clique in a k-tree, then G is also a k-tree. Trees, connected acyclic graphs, are k-trees for k = 1. We introduce a new parameter known as the shell of a k-tree, and from the shell special subclasses of k-trees, tree-like k-trees, are classified. Tree-like k-trees are generalizations of paths, maximal outerplanar graphs, and chordal planar graphs with toughness exceeding one. Let fs = fs( G) be the number of independent sets of cardinality s of G. Then the polynomial I(G; x) = [special characters omitted] fs(G)x s is called the independence polynomial. All rational roots of the independence polynomials of paths are found, and the exact paths whose independence polynomials have these roots are characterized. Additionally trees are characterized that have ?1/q as a root of their independence polynomials for 1 ? q ? 4. The well known vertex and edge reduction identities for independence polynomials are generalized, and the independence polynomials of k-trees are investigated. Additionally, sharp upper and lower bounds for fs of maximal outerplanar graphs, i.e. tree-like 2-trees, are shown along with characterizations of the unique maximal outerplanar graphs that obtain these bounds respectively. These results are extensions of the works of Wingard, Song et al., and Alameddine. Let M1 and M2 be the first and second Zagreb index respectively. Then the minimum and maximum M1 and M2 values for k-trees are determined, and the unique k-trees that obtain these minimum and maximum values respectively are characterized. Additionally, the Zagreb indices of tree-like k-trees are investigated
Higher order assortativity in complex networks
Assortativity was first introduced by Newman and has been extensively studied
and applied to many real world networked systems since then. Assortativity is a
graph metrics and describes the tendency of high degree nodes to be directly
connected to high degree nodes and low degree nodes to low degree nodes. It can
be interpreted as a first order measure of the connection between nodes, i.e.
the first autocorrelation of the degree-degree vector. Even though
assortativity has been used so extensively, to the author's knowledge, no
attempt has been made to extend it theoretically. This is the scope of our
paper. We will introduce higher order assortativity by extending the Newman
index based on a suitable choice of the matrix driving the connections. Higher
order assortativity will be defined for paths, shortest paths, random walks of
a given time length, connecting any couple of nodes. The Newman assortativity
is achieved for each of these measures when the matrix is the adjacency matrix,
or, in other words, the correlation is of order 1. Our higher order
assortativity indexes can be used for describing a variety of real networks,
help discriminating networks having the same Newman index and may reveal new
topological network features.Comment: 24 pages, 16 figure
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