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

On Topological Indices And Domination Numbers Of Graphs

Topological indices and dominating problems are popular topics in Graph Theory. There are various topological indices such as degree-based topological indices, distance-based topological indices and counting related topological indices et al. These topological indices correlate certain physicochemical properties such as boiling point, stability of chemical compounds. The concepts of domination number and independent domination number, introduced from the mid-1860s, are very fundamental in Graph Theory. In this dissertation, we provide new theoretical results on these two topics. We study k-trees and cactus graphs with the sharp upper and lower bounds of the degree-based topological indices(Multiplicative Zagreb indices). The extremal cacti with a distance-based topological index (PI index) are explored. Furthermore, we provide the extremal graphs with these corresponding topological indices. We establish and verify a proposed conjecture for the relationship between the domination number and independent domination number. The corresponding counterexamples and the graphs achieving the extremal bounds are given as well

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group

OLEIC ACID VESICLES: FORMATION, MECHANISMS OF REACTIVITY, AND USES IN DETERMINATION OF TERPENE ACTIVITY

This dissertation will focus on the volatile compounds released upon the burning of incense which are numerous and varied. The first part of this dissertation is the gas chromatography-mass spectral (GC-MS) analysis of burning incense collected via solid phase microextraction (SPME) with the aim of developing a library of compounds found in incense as used in the Orthodox church. The second part of this dissertation has the aim of developing a method for forming oleic acid bilayer vesicle membranes and a fluorescence spectroscopy method by which the reactivities of these vesicles can be analyzed. These reactivities include permeability, fluidity, aggregation, and fusion of the membranes. One family of the volatile compounds found in incense are the terpenes and terpenoids. The reactivity of the terpenes and terpenoids found in incense will be analyzed using the oleic acid vesicles with the hypothesis that terpenes of the same structural groups will act similarly on oleic acid vesicle membranes and these reactivities can be related to mechanistic interactions

Synthesis and characterizations of new organic molecules luminescent for applications in lighting and bioimaging

In recent years, devices based on thermally activated delayed fluorescence (TADF) emitters have proven to be the most promising and efficient approach to convert dark triplet states into emissive singlet states. TADF materials are characterized by a small energy gap between the excited singlet state (S1) and the excited triplet state (T1), this gap called ΔEST is generally less than 0.2 eV. As with phosphorescent emitters, purely organic TADF emitters can recruit excitons from both singlet and triplet states for light emission and thus allow to construct OLED devices reaching 100% internal quantum efficiencies (IQE). An important advantage of these emitters is related with their organic molecular skeleton, thus avoiding the problems associated with the use of heavy metal-based organometallic complexes. Thanks to their unique photophysical properties, organic TADF materials are being studied for a wide range of domains, such as in organic light emitting diodes (OLEDs), solar cells, photocatalysis, etc. Furthermore, TADF-based materials have a unique advantage in some imaging techniques due to their longer life than promptly fluorescent materials. These organic compounds are particularly promising in bioimaging applications due to their low cytotoxicity compared to traditional compounds such as lanthanide complexes. In the course of this thesis, the synthesis and photophysical characterization of new organic TADF molecules was carried out, specifically new coumarin and spiro structures, candidates for the construction of new OLED and as non-toxic biomarkers with efficient emissions in the blue-green spectral region