Total Π-electron Energy Research Articles

{X,Y}-Cyclacene Graphs with Next Nearest Neighbor Interactions

ABSTRACTEigensolutions of {X( = C,B,N),Y( = C,B,N)}-cyclacene graphs with next nearest neighbor (nnn) interactions have been obtained in analytical forms by adapting n-fold rotational symmetry followed by two-fold rotational symmetry (or a plane of symmetry). Expressions of eigensolution indicate the subspectral relationship among such cyclacenes with an even number of hexagonal rings e.g., eigenvalues of {X,Y}-di-cyclacene are found in the eigenspectra of all such even cyclacenes. Total π-electron energies and highest occupied molecular orbital and lowest unoccupied molecular orbital (HOMO–LUMO) gaps are calculated using the analytical expressions obtained and are found to vary negligibly with the variation of nnn interactions in such cyclacenes. Total π-electron energy is found to increase due to increase in restriction intensity of nnn interactions, whereas the HOMO–LUMO gap of polyacenecs having the even number of hexagonal rings and with one electron at each site (atom) decreases with increase in the restriction intensity since such systems contain degenerate half-filled HOMO (bonding or nonbonding) that are much more vulnerable for perturbations imposed through nnn interactions.

Computing the forgotten topological index of four operations on graphs

For a (molecular) graph, the first Zagreb index M1 is equal to the sum of squares of the degrees of vertices, and the second Zagreb index M2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. The F-index of a graph G denoted by F(G) or M3(G) is defined as the sum of cubes of the degrees of vertices of the graph. The total π-electron energy depends on the degree based sum M1(G)=∑v∈VdegG(v)2 and F(G)=∑v∈VdegG(v)3, it was shown in the study of structure-dependency of total π-electron energy in 1972. The first index was named first Zagreb index and the second sum ∑v∈VdegG(v)3 has been never further studied. Recently, this sum was named Forgotten index or the F-index by Furtula and Gutman and it was shown to have an exceptional applicative potential.The first and second Zagreb indices for the four operations on graphs were studied by Deng et al. (2016). In this paper, we extend this study to the F-index of graphs and determine the closed formulas for the F-index of four operations on graphs.

Extremal unicyclic and bicyclic graphs with respect to the [formula omitted]-index

In the study of structure-dependency of the total π-electron energy in 1972, it was shown that this energy depends on the degree based sum ∑v∈VdegG(v)2 and ∑v∈VdegG(v)3, where deg(v) is the degree of a vertex of under consideration molecular graph G. Later on, the sum ∑v∈VdegG(v)2 was named as first Zagreb index. An extensive research work has been done on the Zagreb index and its variants and generalizations, and it become the most investigated topological index in chemical graph theory. But the sum ∑v∈VdegG(v)3 has been never further investigated, except only it was used in finding some of the general first Zagreb index and the zeroth-order general Randić index. Recently, this sum was renamed Forgotten index or F-index and it was shown to have an exceptional applicative potential. In this paper, we determine the extremal graphs with respect to the F-index among the classes of connected unicyclic and bicyclic graphs.

Total π-electron and HOMO energy

A relation is obtained between the total π-electron energy Eπ and the HOMO energy EHOMO, valid within the HMO approximation. This seems to be the very first relation between Eπ and EHOMO ever established. It enables a much more accurate assessment of Eπ of alternant conjugated hydrocarbons than that based on McClelland's formula and on other (n, m)-type approximate expressions.

On extremal Zagreb indices of trees with given domination number

First (M1) and second (M2) Zagreb indices are graph invariants that originate from chemical researches on total π-electron energy of conjugated molecules. There is a legion of articles dealing with these two indices. This paper presents upper bounds on Zagreb indices of trees in terms of domination number. These are strict bounds, and extremal trees are characterized. In addition, a lower bound for the first Zagreb index of trees with a given domination number is determined and the extremal trees are characterized as well. Finally, using previously known upper bound for Harary index (H) in terms of M1 and M2, a unique tree with given domination number that maximizes H is characterized.

Total π-Electron Energy of Conjugated Molecules with Non-bonding Molecular Orbitals

Abstract Lower and upper bounds for the total π-electron energy are obtained, which are applicable to conjugated π-electron systems with non-bonding molecular orbitals (NBMOs). These improve the earlier estimates, in which the number of NBMOs has not been taken into account.

The asymptotic value of the zeroth-order Randić index and sum-connectivity index for trees

The zeroth-order Randić index and sum–connectivity index are two indices based on the vertex degrees. They appeared in the topological formula for the total π-electron energy of conjugated molecules and attracted a lot of attention in recent years. Let Tn be the set of trees of order n. Suppose each tree in Tn is equally likely. We get that for almost every tree, the zeroth-order Randić index is among (r1 ± ε)n and the sum–connectivity index is among (r2 ± ε)n, where r1, r2 are some constants and ε is any positive real number.

有向双圈图的第二小斜能量

斜能量在化学能量方面具有广泛的应用，由于共轭分子的量子化学的一个重要特性是它的π-电子能量。在本篇文献中，我们通过比较有向双圈图的斜特征多项式的系数，给出有向双圈图的斜能量的偏序关系，从而得到有向双圈图的第二小斜能量。 The energy of a graph has closed links to chemistry, since an important quantum-chemical cha-racteristic of a conjugated molecule is its total π-electron energy. In this paper, we compared the characteristic polynomial coefficients of oriented bicyclic graphs and gave the skew energy’s partial relation of oriented bicyclic graphs, and thus we got the second minimal skew energy of oriented bicyclic graphs.

Eigensolutions of cyclopolyacene graphs

A graph of {X, Y}-cyclopolyacene with n of hexagonal rings has been presented that contains four orbits, of which orbits 1 and 4 are occupied by the X-type of vertex and orbits 2 and 4 are occupied by the Y-type, or vice versa. Eigensolutions for such a graph have been derived in analytical form through the use of rotational symmetry followed by a plane of symmetry. Varying X ( = C, N, B, …) and Y ( = C, N, B, …) several types of cyclopolyacene graph may be obtained. Eigenvalue-expressions for such systems containing C, N and B have been shown in analytical form and their total π-electron energies with 2–6 hexagonal rings have been calculated with the help of the expressions developed.

On the minimal energy of graphs

The energy of a graph is the sum of the absolute values of the eigenvalues of the graph which is used to approximate the total π-electron energy of the molecule. In this paper, we determine the (n,e)-graphs with minimal energy for e=n+1 and n+2, which is giving a complete solution to the conjecture for e=n+1 and e=n+2 proposed by Caporossi et al. in [4]. Moreover, we determine the graphs with the minimal and second-minimal energies for n−1≤e≤3n2−3, and the unique graph with minimal energy for 3n−52≤e≤2n−4 among all quasi-trees with n vertices and e edges, respectively.

Eigensolutions of dodecahedron graphs

Eigensolutions of 20-vertex cage (i.e. dodecahedron) have been determined with the use of fivefold rotational symmetry. For homo-dodecahedron the eigensolutions become analytical but for the hetero-dodecahedron having two different types of atoms ((C,N),(C,B),(B,N)) the eigensolutions are found to be factored out into five 4-degree polynomials with one corresponding to nondegenerate and other four corresponding to two degenerate eigensolutions. Eigenspectra and total π-electron energies of homo- and hetero-dodecahedron graphs have been calculated.

New bounds and approximations for total π-electron energy: A critical test

The quality and correlating ability of some recently deduced bounds and approximate formulas for total π-electron energy are tested on a sample of 106 Kekuléan benzenoid hydrocarbons. It was found that not a single new approximate formula and not a single new bound is better than those from the 1970s.

Comparing energy and Randic energy

The recently conceived Randić energy (RE) is examined, and its relation to the (earlier much studied) total π-electron energy (E) is investigated. Within classes of molecular graphs, there exists a relatively good (increasing) linear correlation between RE and E. However, several significant differences between the structure-dependencies of RE and E have been discovered, the most striking of which is their dependence on the number m of edges of the underlying graph. Whereas, with increasing m, the average value of E increases, reaches a maximum and then decreases, the average value of RE monotonically decreases. The structure of the connected graph with a fixed number of vertices and maximal RE value was established.

Bounds for all graph energies

In addition to the (ordinary) graph energy, which is directly related to the total π-electron energy, scores of other graph energies have been introduced in the last few years. We derive lower and upper bounds, applicable to any of these graph energies.

Generalizing the McClelland Bounds for Total π-Electron Energy

In 1971 McClelland obtained lower and upper bounds for the total π-electron energy. We now formulate the generalized version of these bounds, applicable to the energy-like expression EX = Σn i =1 |xi − x̅|, where x1,x2, . . . ,xn are any real numbers, and x̅ is their arithmetic mean. In particular, if x1,x2, . . . ,xn are the eigenvalues of the adjacency, Laplacian, or distance matrix of some graph G, then EX is the graph energy, Laplacian energy, or distance energy, respectively, of G.

Sharp upper bounds for total π-electron energy of alternant hydrocarbons

The energy E(G) of a graph G is defined as the sum of the absolute values of all the eigenvalues of the adjacency matrix of the graph G. This quantity is used in chemistry to approximate the total π-electron energy of molecules and in particular, in case G is bipartite, alternant hydrocarbons. In this paper, we show that if G = (V1, V2; E) is a bipartite graph with m n1 edges and |V1 |= n1 n2 =| V2| ,t hen E(G) 2m √ n1n2 + 2 (n2 − 1) m − m 2 n1n2 and E(G) √ n1n2(1 + √ n2)

Total π-electron energy and Laplacian energy: How far the analogy goes?

The Laplacian energy LE is a newly introduced molecular-graph-based analog of the total ?-electron energy E. It is shown that LE and E have a similar structure-dependency only when molecules of different sizes are compared, when a good linear correlation between them exists. Within classes of isomers, LE and E are either not correlated at all or (as in the case of acyclic systems) are inversely proportional. The acyclic graphs and molecular graphs having the greatest and smallest LE values (determined in this work) differ significantly from those (previously known) having the greatest and smallest E values.

Double hexagonal chains with minimal total π-electron energy

The total energy of all π-electrons in a conjugated hydrocarbon (within the framework of HMO approximation) is the sum of the absolute value of all the eigenvalues of its corresponding graph. In this paper, we consider “double hexagonal chains” as benzenoids constructed by successive fusions of successive naphthalenes along a zig–zag sequence of triples of edges as appear on opposite sides of each naphthalene unit. It is shown that if the fusions are such as to give a polyaceacene then the total π-electron energy is the minimum from among all the double hexagonal chains with the same number of naphthalene units.

Relating Total π-Electron Energy and Resonance Energy of Benzenoid Molecules with Kekulé- and Clar-Structure-Based Parameters

Within classes of isomeric benzenoid hydrocarbons various Kekule- and Clar-structure-based parameters (Kekule structure count, Clar cover count, Herndon number, Zhang–Zhang polynomial) are all mutually correlated. This explains why both the total π-electron energy (E), the Dewar resonance energy (DRE), and the topological resonance energy (TRE) are well correlated with all these parameters. Nevertheless, there exists an optimal value of the variable of the Zhang–Zhang polynomial for which it yields the best results. This optimal value is negative-valued for E, around zero for TRE, and positive-valued for DRE. A somewhat surprising result is that TRE and DRE considerably differ in their dependence on Kekule- and Clar-structure-based parameters.

On the Relationship between π-Electron Energy and Topological Resonance Energy

Within series of isomeric benzenoid hydrocarbons there is a very good linear correlation between the topological resonance energy (TRE) and the total π-electron energy (Eπ ). Furthermore, the slope of the TRE vs. Eπ regression line is almost independent of the benzenoid isomers considered, and (for all sets of isomers) is nearly equal to 0.5. This implies that comparison of benzenoid isomers with regard to their aromaticity can be made, with equal success, by using both TRE and Eπ . However, Eπ is computed significantly simpler than TRE, and thus advantage should be given to the former. Correlations between TRE and Eπ exist also in the case of non-benzenoid isomers (both alternant and non-alternant), but are of much inferior quality.