Zagreb Connection Indices of Molecular Graphs Based on Operations

Topological index (numeric number) is a mathematical coding of the molecular graphs that predicts the physicochemical, biological, toxicological, and structural properties of the chemical compounds that are directly associated with the molecular graphs. The Zagreb connection indices are one of the TIs of the molecular graphs depending upon the connection number (degree of vertices at distance two) appeared in 1972 to compute the total electron energy of the alternant hydrocarbons. But after that, for a long period, these are not studied by researchers. Recently, Ali and Trinajstic Mol. Inform. 372018,1−7 restudied the Zagreb connection indices and reported that the Zagreb connection indices comparatively to the classical Zagreb indices provide the better absolute value of the correlation coefficient for the thirteen physicochemical properties of the octane isomers (all these tested values have been taken from the website http://www.moleculardescriptors.eu). In this paper, we compute the general results in the form of exact formulae & upper bounds of the second Zagreb connection index and modified first Zagreb connection index for the resultant graphs which are obtained by applying operations of corona, Cartesian, and lexicographic product. At the end, some applications of the obtained results for particular chemical structures such as alkanes, cycloalkanes, linear polynomial chain, carbon nanotubes, fence, and closed fence are presented. In addition, a comparison between exact and computed values of the aforesaid Zagreb indices is also included.

Zagreb Connection Indices of Subdivision and Semi-Total Point Operations on Graphs

Representation or coding of the molecular graphs with the help of numerical numbers plays a vital role in the studies of physicochemical and structural properties of the chemical compounds that are involved in the molecular graphs. For the first time, the modified first Zagreb connection index appeared in the paper by Gutman and Trinajstic (1972) to compute total electron energy of the alternant hydrocarbons, but after that, for a long time, it has not been studied. Recently, Ali and Trinajstic (2018) restudied the first Zagreb connection index ZC1, the second Zagreb connection index ZC2, and the modified first Zagreb connection index ZC1∗ to find entropy and acentric factor of the octane isomers. They also reported that the values provided by the International Academy of Mathematical Chemistry show better chemical capability of the Zagreb connection indices than the ordinary Zagreb indices. Assume that S1 and S2 denote the operations of subdivision and semitotal point, respectively. Then, the S-sum graphs Q1+QS2 are obtained by the cartesian product of SQ1 and Q2, where S∈S1,S2, Q1andQ2 are any connected graphs, and SQ1 is a graph obtained after applying the operation S on Q1. In this paper, we compute the Zagreb connection indices (ZC1, ZC2, and ZC1∗) of the S-sum graphs in terms of various topological indices of their factor graphs. At the end, as an application of the computed results, the Zagreb connection indices of the S-sum graphs obtained by the particular classes of alkanes are also included.

Tripartite graphs with energy aggregation

The aggregate of the absolute values of the graph eigenvalues is called the energy of a graph. It is used to approximate the total _-electron energy of molecules. Thus, finding a new mechanism to calculate the total energy of some graphs is a challenge; it has received a lot of research attention. We study the eigenvalues of a complete tripartite graph Ti,i,n−2i , for n _ 4, based on the adjacency, Laplacian, and signless Laplacian matrices. In terms of the degree sequence, the extreme eigenvalues of the irregular graphs energy are found to characterize the component with the maximum energy. The chemical HMO approach is particularly successful in the case of the total _-electron energy. We showed that some chemical components are equienergetic with the tripartite graph. This discovering helps easily to derive the HMO for most of these components despite their different structures.

Nature ofE2X2σ(4c–6e) of theX---E—E---Xtype at naphthalene 1,8-positions and model, elucidated by X-ray crystallographic analysis and QC calculations with the QTAIM approach

The nature ofE2X2σ(4c–6e) of theX-*-E-*-E-*-Xtype is elucidated for 1-(8-XC10H6)E–E(C10H6X-8′)-1′ [(1)E,X= S, Cl; (2) S, Br; (3) Se, Cl; (4) Se, Br] after structural determination of (1), (3) and (4), together with modelA[MeX---E(H)—E(H)---XMe (E= S and Se;X= Cl and Br)]. The quantum theory of atoms-in-molecules dual functional analysis (QTAIM-DFA) is applied. The total electron energy densitiesHb(rc) are plottedversus Hb(rc) –Vb(rc)/2 for the interactions at the bond critical points (BCPs; *), whereVb(rc) show the potential energy densities at the BCPs. Data for the perturbed structures around the fully optimized structures are employed for the plots, in addition to those of the fully optimized structures. The plots were analysed using the polar coordinate (R, θ) representation of the data of the fully optimized structures. Data containing the perturbed structures were analysed by (θp, κp), where θpcorresponds to the tangent line of the plot and κpis the curvature. Whereas (R, θ) shows the static nature, (θp, κp) represents the dynamic nature of interactions.E-*-Eare all classified as shared shell (S) interactions for (1)–(4) and as weak covalent (Cov-w) in nature (S/Cov-w). The nature ofpureCS (closed shell)/typical-HB (hydrogen bond) with no covalency is predicted forE-*-Xin (1) and (3),regularCS/typical-HB nature with covalency is predicted for (4), and an intermediate nature is predicted for (2). The NBO energies evaluated forE-*-Xin (1)–(4) are substantially larger than those in modelAdue the shortened length at the naphthalene 1,8-positions. The nature ofE2X2of σ(4c–6e) is well elucidatedviaQTAIM-DFA.

A graph theoretical approach to cis/trans isomerism

A simple graph-theory-based model is put forward, by means of which it is possible to express the energy difference between geometrically non-equivalent forms of a conjugated polyene. This is achieved by modifying the adjacency matrix of the molecular graph, and including into it information on cis/trans constellations. The total ?-electron energy thus calculated is in excellent agreement with the enthalpies of the underlying isomers and conformers.

Calculations and experimental investigation of pulse transmission system in the typical module of the facility “Gamma”

For the last few years in INRP RFNC-VNIIEF the works on development of a multi-module «Gamma» facility have been conducted. An important part of each module is a pulse transmission system (PTS), providing transportation of a high-volt electromagnetic pulse (~2.3 MV, ~60 ns) to a diode load, positioned at an angle of ~80° to the axis of a module's forming system. Basic PTS units: a water-insulated transmission line (WTL), having a bended section, a vacuum insulator stack and a magnetically-insulated transmission line (MITL). At the first stage an experimental sample of PTS with diameter 0.65 m was studied. Performed studies allowed a conclusion that the given experimental PTS sample did not possess enough electric strength, what was a reason for electric breakdowns in the bended section of WTL. Reasons for breakdown occurrence were analyzed; conclusions were made on the necessity for increasing PTS diameter. As a result a PTS version with diameter ~1 m was developed. This paper presents results of the experimental studies as a part of the facility module. Totally 200 shots of the module were performed with given PTS at different charge voltage of its forming lines. Reliable and steady operation of all PTS units, as well as correspondence between output module parameters and their calculated values were proved. When using PTS, without MITL in the module diode load, with impedance ~3 Ohm the pulses with power 1.5 TW and total electron energy in a pulse ~80 kJ were obtained. When using PTS with cylindrical MITL of 1.6 m length, the pulse power was ~1.4 TW.

Estimating the total π-electron energy

The paper gives a short survey of the most important lower and upper bounds for total ?-electron energy, i.e., graph energy (E). In addition, a new lower and a new upper bound for E are deduced, valid for general molecular graphs. The strengthened versions of these estimates, valid for alternant conjugated hydrocarbons, are also reported.

CHANNELING EXPERIMENTS WITH ELECTRONS AT THE MAINZ MICROTRON MAMI

The dechanneling process of electrons in silicon single crystals has been studied at the Mainz Microtron MAMI for (110)-planar channeling of electrons at beam energies between 195 and 855 MeV. Dechanneling lengths were derived from a high and a low energy loss signal of the electrons which were recorded as function of the crystal orientation with respect to the beam direction for various crystal thicknesses in the range between 14.7 µm and 467 µm. The high energy loss signal corresponds to an energy loss of about 75 % of the total electron energy by emission of a bremsstrahlung photon, while the low energy signal to an energy loss of 0.7-1.7 % by emission of channelling radiation. While the high energy signal saturates as function of the crystal thickness, the low energy signal does not. The nearly constant dechanneling length of about 35 µm, as extracted from the high energy signal, is interpreted to originate from a small fraction of electron which initially occupy deeply bound quantum states.

Stability order of isomeric benzenoid hydrocarbons and Kekulé structure count

The commonly accepted opinion that the thermodynamic stability of isomeric benzenoid hydrocarbons (assessed by their total ?-electron energy and various resonance energies) increases with increasing number of Kekul? structures is shown to be violated in numerous cases. The smallest examples of such anomalous behavior are two hexacyclic pericondensed benzenoids of formula C24H14 and several pairs of heptacyclic catacondensed benzenoids of formula C30H18.