Some Less Familiar Properties of Randić Index

Several mathematically relevant properties of the Randić connectivity index, that may be less familiar to the chemical community, are outlined and commented. This work is licensed under a Creative Commons Attribution 4.0 International License

UPPER AND LOWER BOUNDS FOR THE SKEW HERMITIAN RANDIC ENERGY OF STANDARD GRAPHS

Abstract Let us consider a simple graph . The energy of the graph is defined as the sum of the absolute values of the eigen values of the adjacency matrix [1] & [2]. The energy of , denoted by , is called Skew- Hermitian Randić energy, which is defined as the sum of the absolute values of its eigenvalues of , that is, [9]. The total π electron energy of conjugated hydrocarbon molecules are closely connected with graph invariant[10],[11]. Recently based on the eigen values of graph matrices various energies are computed. For a graph matrix, we can determine the eigen values based on which we can compute the energy of the graph. In this paper, we have determined the Skew-Hermitian Randic Energy of some standard graphs[13],[14],[15]

Complexity of Some Interesting (Chemical) Graphs

Complexity of some interesting polycyclic graphs is expressed in terms of the corresponding spanning trees. Graphs considered were a selection of all connected graphs with four and five vertices, graphs composed of two parts, or more parts, connected by a single edge, the Petersen graph, the Blanuša graph, the Desargues-Levy graph and the Schlegel graph of buckminsterfullerene

The Total π-Electron Energy Saga

The total π-electron energy, as calculated within the Hückel tight-binding molecular orbital approximation, is a quantum-theoretical characteristic of conjugated molecules that has been conceived as early as in the 1930s. In 1978, a minor modification of the definition of total π-electron energy was put forward, that made this quantity interesting and attractive to mathematical investigations. The concept of graph energy, introduced in 1978, became an extensively studied graph-theoretical topic, with many hundreds of published papers. A great variety of graph energies is being considered in the current mathematical-chemistry and mathematical literature. Recently, some unexpected applications of these graph energies were discovered, in biology, medicine, and image processing. We provide historic, bibliographic, and statistical data on the research on total π-electron energy and graph energies, and outline its present state of art. The goal of this survey is to provide, for the first time, an as-complete-as-possible list of various existing variants of graph energy, and thus help the readers to avoid getting lost in the jungle of references on this topic. This work is licensed under a Creative Commons Attribution 4.0 International License

The Total π-Electron Energy Saga

The total π-electron energy, as calculated within the Hückel tight-binding molecular orbital approximation, is a quantum-theoretical characteristic of conjugated molecules that has been conceived as early as in the 1930s. In 1978, a minor modification of the definition of total π-electron energy was put forward, that made this quantity interesting and attractive to mathematical investigations. The concept of graph energy, introduced in 1978, became an extensively studied graph-theoretical topic, with many hundreds of published papers. A great variety of graph energies is being considered in the current mathematical-chemistry and mathematical literature. Recently, some unexpected applications of these graph energies were discovered, in biology, medicine, and image processing. We provide historic, bibliographic, and statistical data on the research on total π-electron energy and graph energies, and outline its present state of art. The goal of this survey is to provide, for the first time, an as-complete-as-possible list of various existing variants of graph energy, and thus help the readers to avoid getting lost in the jungle of references on this topic. This work is licensed under a Creative Commons Attribution 4.0 International License

Graphs and networks theory

This chapter discusses graphs and networks theory

An edge-based formulation of elastic network models

We present an edge-based framework for the study of geometric elastic network models to model mechanical interactions in physical systems. We use a formulation in the edge space, instead of the usual node-centric approach, to characterise edge fluctuations of geometric networks defined in d- dimensional space and define the edge mechanical embeddedness, an edge mechanical susceptibility measuring the force felt on each edge given a force applied on the whole system. We further show that this formulation can be directly related to the infinitesimal rigidity of the network, which additionally permits three- and four-centre forces to be included in the network description. We exemplify the approach in protein systems, at both the residue and atomistic levels of description

A Dual of the Cycle Theorem and its Application to Molecular Complexity

The duality alluded to in the title is that between the faces and vertices of a graph embedded on a surface. Its recognition in the context of the five Platonic solids is classic. Algebraically, it is present in the equation for Euler’s Polyhedron Theorem and in the various extensions thereof. The Cycle Theorem (CT) establishes a formula for the number of spanning trees contained in a graph embedded on a surface. It is based on the mutual incidences of its cycles (circuits which also carry a sense of direction), i.e., of sub-graphs of the Cn type, endorsed with a sense. These appear (though not exclusively) as the boundaries of faces, so that, so to speak, the Cycle Theorem establishes a result which is essentially about vertices via relations between faces. Among several possible duals of the Cycle Theorem there might thus be one that estab¬lishes a relation which is essentially about faces via relations between vertices. In order to formulate one, we define, for an embedded graph, a feature concerning faces that is dual to a spanning tree. We call it a ladder. A formula is presented for the number of ladders contained in a graph which, in some cases, introduces the concept of ‘artificial vertices’. It is based on the mutual incidences of its vertices. Its form is clearly analogous, or ‘dual’, to the Cycle Theorem formula for spanning trees, previously proposed (in this journal — 2004) by three of the present authors, together with Klein and Sachs. A new index is proposed, which involves ladders. We call it the Patency Index of a graph; its numerical value may be related to molecular complexity. It is effectively the dual of, and is entirely analogous to, the Spanning-Tree Density Index which was earlier (2003) proposed, defined and applied to molecular graphs by one of the present authors and Trinajstić. This work is licensed under a Creative Commons Attribution 4.0 International License

Onečišćenja u ljekovitom bilju i biljnim proizvodima

Medicinal plants have a long history of use in therapy throughout the world and still make an important part of traditional medicine. Thus, medicinal plants and herbal products must be safe for the patient (consumer). This review addresses biological contaminants (microbes and other organisms) and chemical contaminants (mycotoxins, toxic elements such as heavy metals, and pesticide residues) as major common contaminants of medicinal herbs and herbal products. To prevent and screen for contamination and ensure safety and conformity to quality standards, medicinal herbs and herbal products should be included in appropriate regulatory framework.Ljekovito bilje i biljni proizvodi već tisućljećima nalaze široku primjenu u različitim sustavima tradicionalnog liječenja. Stoga je njihova neškodljivost, ponajprije uvjetovana kakvoćom biljne sirovine, od izuzetne važnosti za javno zdravstvo. Od mogućih čimbenika koji utječu na kakvoću ljekovitog bilja i biljnih proizvoda ovaj pregledni rad osvrće se na najčešće prisutna biološka (mikroorganizmi) i kemijska onečišćenja (mikotoksini, toksični elementi poput teških metala te ostaci pesticida). S ciljem postizanja ujednačenih standarda kakvoće te osiguranja sigurnosti primjene biljnih proizvoda od vitalne su važnosti zakonski propisi koji moraju u odgovarajućim regulatornim okvirima obuhvatiti ovu skupinu proizvoda s naglaskom na sprječavanju i ispitivanju njihovih mogućih onečišćenja