Total Π-electron Energy Research Articles

Data-driven approaches to study the spectral properties of chemical structures

The molecular energy, which is the sum of all eigenvalues, is crucial in determining the total π-electron energy of conjugated hydrocarbon molecules. We used machine learning techniques to calculate the energy, inertia, nullity, signature, and Estrada index of molecular graphs for bismuth tri-iodide and benzene rings embedded in P-type surfaces within 2D networks. We applied MATLAB to extract the actual eigenvalues from the data and developed general equations for these molecular properties. We then used these equations to estimate the values and compared them to the actual values through graphical analysis. Our results demonstrate the potential of data-driven techniques in predicting molecular properties and enhancing our understanding of spectral theory.

Comparative analysis of temperature-based graphical indices for correlating the total π-electron energy of benzenoid hydrocarbons

In a graph [Formula: see text], the temperature [Formula: see text] of a vertex [Formula: see text] is defined as [Formula: see text], where n is the order of G and [Formula: see text] is the valency/degree of x. A topological/graphical index [Formula: see text] is a map [Formula: see text], where ∑ (respectively, [Formula: see text]) is the set of simple connected graphs (respectively, real numbers). Graphical indices are employed in quantitative structure-property relationship (QSPR) modeling to predict physicochemical/thermodynamic/biological characteristics of a compound. A temperature-based graphical index of a chemical graph G is defined as [Formula: see text], where [Formula: see text] is a symmetric 2-variable map. In this paper, we introduce two new novel temperature-based indices named as the reduced reciprocal product-connectivity temperature ([Formula: see text]) index and the geometric-arithmetic temperature ([Formula: see text]) index. The predictive potential of these indices has been investigated by employing them in structure-property modeling of the total [Formula: see text]-electronic energy [Formula: see text] of benzenoid hydrocarbons. In order to validate the statistical inference, the lower 30 BHs have been opted as test molecules as their experimental data for [Formula: see text] is also publicly available. First, we employ a computer-based computational method to compute temperature indices of 30 lower BHs. Certain QPSR models are proposed by utilizing the experimental data of [Formula: see text] for the BHs. Our statistical analysis suggests that the most efficient regression models are, in fact, linear. Our statistical analysis asserts that both [Formula: see text] and [Formula: see text] outperformed all the existing temperature indices for correlating [Formula: see text] for the BHs. The results suggest their further employability in QSPR modeling. Importantly, our research contributes toward countering proliferation of graphical indices.

QSPR analysis of carbon allotropes by employing molecular descriptors and information entropies

The inherent qualities of carbon nanosheets that come from their underlying molecular structure have drawn an enormous amount of interest from researchers. Molecular descriptors are commonly used graph-theoretic metrics to analyze the physicochemical properties of a molecule based on its molecular structure. Our current research validates the use of molecular descriptors in studying three recently developed carbon allotropes: pentagraphene, phagraphene, and phographene. These allotropes have significant thermal, dynamic, and mechanical stabilities that are comparable to graphene. General analytical expressions for both degree and neighborhood degree sum-based indices of these nanosheets are derived through the utilization of graph theoretical methods. Following a filtering process of the significant indices, regression models that accurately predict atomic density, specific heat capacity and total π-electron energy of carbon allotropes are developed. Additionally, Shannon entropy is introduced as an information index, which can be utilized for predictive studies in the future and as a tool for comparing structural complexity by incorporating topological index in its definition.

Extremal hyper-Zagreb index of trees of given segments with applications to regression modeling in QSPR studies

Introduced by Gutman & Trinajstić in 1972, the two well-known Zagreb indices efficiently determine the total π-electron energy of alternant hydrocarbons. The hyper-Zagreb index is a variant of the classical first Zagreb index. A tree is an acyclic graphical structure. A segment in a tree is equivalent to a vertex of degree two. This paper derives sharp upper and lower bounds on the hyper-Zagreb index of trees with given number of segments. We also characterize the corresponding trees achieving those bounds. The second part of the paper studies the predictive potential of the hyper-Zagreb index for determining physicochemical properties such as the enthalpies of combustion, formation, sublimation, and vaporization of monocarboxylic acids. Our statistical analysis asserts that the hyper-Zagreb index correlates these physicochemical properties of monocarboxylic acids with correlation coefficient ρ>0.99. The strongest correlation by the hyper-Zagreb index was obtained with the enthalpy of combustion having correlation coefficient ρ=0.999991. This warrants the further employability of the hyper-Zagreb index in QSPR modeling. This potential applicability justifies the construction of the hyper-Zagreb graphical index whose motivation of defining was purely mathematical.

On the spectrum, energy and Laplacian energy of graphs with self-loops

The total energy of a conjugated molecule's π-electrons is a quantum-theoretical feature that has been known since the 1930s. It is determined using the Huckel tight-binding molecular orbital (HMO) method. In 1978, a modified definition of the total π-electron energy was introduced, which is now known as the graph energy. It is calculated by summing the absolute values of the eigenvalues of the adjacency matrix. Quiet Recently in the year 2022, Gutman extended the concept of conjugated systems to hetero-conjugated systems which is the extension of ordinary graph energy to energy of graph with self loops. Let Gσ be an order (vertices) ‘p’ graph with ‘q’ edges and σ− self loops. The adjacency matrix of Gσ is defined by A(Gσ)=(aij) if vi∼adjvj then aij=1; if vi=vj where vi∈Vσ then aii=1 and zero otherwise, where Vσ represents the set of all vertices with loops. Then the energy of graph with self loops is defined as E(Gσ)=∑|λi−σ/p|. In this paper, we aim to analyze the adjacency and Laplacian spectra of certain non-simple standard graphs that contain self-loops. We also calculate the energy and Laplacian energy of these graphs with loops. Furthermore, we derive lower bounds for the energy of any graph containing loops and develop a MATLAB algorithm to calculate these quantities for selected non-simple standard graphs with self-loops. Our study evaluates the strength of a graph by considering the presence of loops, which are edges that connect a vertex to itself. This approach accounts for the impact of each vertex on the entire structure of the graph. By analyzing the energy of a graph with loops, we can gain a better understanding of its distinctive characteristics and behavior.

Analytical forms of the coefficients of the factored characteristic polynomials for zigzag SWCNT graphs

Factors of characteristic polynomials (CP) of zigzag single walled carbon nanotube (SWCNT) graphs have been constructed in generalized forms using rotational symmetry with respect to the tube axis. The CP coefficients in the factors have been expressed in analytical forms involving derivatives of different powers of pj, where pj=(1+ωj)(1+ωj*), ωj=0,1,2,…,(R−1), are the R-th roots of unity and R is the number of fused hexagonal rings per belt in the SWCNT graph. Some important results regarding total π-electron energy per electron, HOMO-LUMO energy gap and number of Kekulé structures for the SWCNTs have been derived from an analysis of the CPs and graph eigenspectra.

On physical analysis of synthesis strategies and entropy measures of dendrimers

Dendrimers achieved great consideration in gene and drug delivery applications because of having highly administrable architecture. Unambiguous structure of dendrimers might reduce the unpredictability related to the molecule’s shape and size, and also boost the accuracy of drug delivery. Dendrimers have exclusive physical and chemical properties due to which they have extensive range of potential applications like chemical sensors, light harvesting material, enhance the solubility, antitumer therapy, medical diagnostics, drug delivery system, catalysts, and many more. With the total π-electron energy, the degree-based topological indices have a lot of iterations. In this paper, our desideration is to compute the topological aspects of degree based entropy for fractal and cayley tree type dendrimers. More preciously, we explore two tree type dendrimers denoted by Fr ant Cm,n. Moreover, entropies are estimated of these two structures by generating a correlation between degree based topological indices and their entropies.

New Results on the Forgotten Topological Index and Coindex

The ℱ -coindex (forgotten topological coindex) for a simple connected graph G is defined as the sum of the terms ζ G 2 y + ζ G 2 x over all nonadjacent vertex pairs x , y of G , where ζ G y and ζ G x are the degrees of the vertices y and x in G , respectively. The ℱ -index of a graph is defined as the sum of cubes of the vertex degrees of the graph. This was introduced in 1972 in the same paper where the first and second Zagreb indices were introduced to study the structure dependency of total π -electron energy. Therefore, considering the importance of the ℱ -index and ℱ -coindex, in this paper, we study these indices, and we present new bounds for the ℱ -index and ℱ -coindex.

On the Energy of Benzenoid Hydrocarbons

In chemistry, the energy of a graph is of interest since it can be used to approximate the total π-electron energy of molecules. The spectral moments of the edge adjacency matrix and adjacency matrix recently have been successfully employed in quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) studies of alkanes, alkyl halides, benzyl alcohols, cycloalkanes, and benzenoid hydrocarbons. Let G be a molecular graph and L(G) be a line graph, then the eigenvalues of G and L(G) are denoted by, and respectively. The energies A and L(G) are defined as and In this paper, we examined both energies with spectral moments of the edge adjacency matrix and adjacency matrix of benzenoid hydrocarbons. These theoretical conclusions provide practical guiding significance for pharmaceutical engineering, complex network and quantify the degree of folding of long organic molecules.

Bounds of Degree-Based Molecular Descriptors for Generalized F -sum Graphs

A molecular descriptor is a mathematical measure that associates a molecular graph with some real numbers and predicts the various biological, chemical, and structural properties of the underlying molecular graph. Wiener (1947) and Trinjastic and Gutman (1972) used molecular descriptors to find the boiling point of paraffin and total π -electron energy of the molecules, respectively. For molecular graphs, the general sum-connectivity and general Randić are well-studied fundamental topological indices (TIs) which are considered as degree-based molecular descriptors. In this paper, we obtain the bounds of the aforesaid TIs for the generalized F -sum graphs. The foresaid TIs are also obtained for some particular classes of the generalized F -sum graphs as the consequences of the obtained results. At the end, 3 D -graphical presentations are also included to illustrate the results for better understanding.

EXTREMAL F-INDICES FOR BICYCLIC GRAPHS WITH k PENDANT VERTICES

Long back in 1972, it was shown that the sum of the squares of vertex degrees and the sum of cubes of vertex degrees of a molecular graph both have large correlations with total π-electron energy of the molecule. Later on, the sum of squares of vertex degrees was named as first Zagreb index and became one of the most studied molecular graph parameter in the field of chemical graph theory. Whereas, the other sum remained almost unnoticed until recently except for a few occasions. Thus it got the name “forgotten” index or F-index. This paper investigates extremal graphs with respect to F-index among the class of bicyclic graphs with n vertices and k pendant vertices, 0 ≤ k ≤ n - 4. As consequences, we obtain the bicyclic graphs with largest and smallest F-indices.

Computation of Degree-Based Topological Descriptors Using M-Polynomial for Coronoid Systems

This paper aims to compute the degree(valence)-based molecular descriptors for coronoid polycyclic aromatic hydrocarbons. The myriad types of coronoids have potential far-reaching applications in evolving fields such as nanotechnology and synthetic organic chemistry due to their superaromaticity and other extraordinary geometric and electronic properties. The degree-based connectivity descriptors, like the Randić, the sum-connectivity, the first and second Zagreb indices, and its co-indices have a strong correlation with the total π-electron energy and other properties which are significant for the synthesis of coronoid polycyclic aromatic compounds. In this article, the various degree(valence)-based topological descriptors are computed using the M-polynomial for coronoid structures with cavities that vary depending on their topology, which will benefit future researchers in exploring and synthesizing new coronoid structures. The analytical expressions for recently developed molecular descriptors are also derived. The degree-based indices have not been studied earlier for coronoid structures, which contributes substantial weightage to this research work. Additionally, the computed degree-based indices are compared both graphically and numerically.

Bounds on F-index of tricyclic graphs with fixed pendant vertices

Abstract The F-index F(G) of a graph G is obtained by the sum of cubes of the degrees of all the vertices in G. It is defined in the same paper of 1972 where the first and second Zagreb indices are introduced to study the structure-dependency of total π-electron energy. Recently, Furtula and Gutman [J. Math. Chem. 53 (2015), no. 4, 1184–1190] reinvestigated F-index and proved its various properties. A connected graph with order n and size m, such that m = n + 2, is called a tricyclic graph. In this paper, we characterize the extremal graphs and prove the ordering among the different subfamilies of graphs with respect to F-index in $\begin{array}{} \displaystyle {\it\Omega}^{\alpha}_n \end{array}$, where $\begin{array}{} \displaystyle {\it\Omega}^{\alpha}_n \end{array}$ is a complete class of tricyclic graphs with three, four, six and seven cycles, such that each graph has α ≥ 1 pendant vertices and n ≥ 16 + α order. Mainly, we prove the bounds (lower and upper) of F(G), i.e $$\begin{array}{} \displaystyle 8n+12\alpha +76\leq F(G)\leq 8(n-1)-7\alpha + (\alpha+6)^3 ~\mbox{for each}~ G\in {\it\Omega}^{\alpha}_n. \end{array}$$

Two Stability Criteria for Benzenoid Hydrocarbons and Their Relation

A new, simple, relation is established between the total π-electron energy and the HOMO-LUMO gap, applicable to benzenoid hydrocarbons.

On bounds for topological descriptors of φ-sum graphs

The properties of chemical compounds are very important for the studies of the non-isomorphism phenomenon's related to the molecular graphs. Topological indices (TIs) are one of the mathematical tools which are used to study these properties. Gutman and Trinajsti [Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons. Chem Phys Lett. 1972;17(4):535–538] defined the Zagreb indices (descriptors) to find correlation value between a molecular graph and its total π-electron energy. Later on, Bollobás and Erdös [Graphs of extremal weights. Ars Comb; 1998;50:225–233] defined the most general form of these indices (descriptors) called by general Randić index (GRI) and first general Zagreb index (FGZI), respectively. In this paper, we computed the bounds for FGZI and GRI of φ-sum graphs, obtained by the strong product of the graph with another graph Γ, where is constructed using four subdivision operations on the graph G. At the end, we also include the results for some particular families of graphs as the applications of the obtained results.

The Laplacian energy of diameter 4 trees

Let Tn be the set of trees with n vertices and having diameter 4. We show that four classes in Tn satisfy a conjecture posed by Radenkovi and Gutman [Total π-electron energy and Laplacian energy: how far the analogy goes? J. Serb. Chem. Soc., 72:12:1343–1350, 2007]. The conjecture relates the Laplacian energy of path Pn , tree Tn , and star Sn .

Hückel Molecular Orbital Quantities of {X,Y}-Cyclacene Graphs Under Next-Nearest-Neighbour Approximations in Analytical Forms

Abstract Hückel molecular orbital (HMO) quantities, viz., electron densities, charge densities, bond orders, free valences, total π-electron energies and highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO–LUMO) or band gaps of {X,Y}-cyclacene graphs under next-nearest-neighbour (nnn) approximations are expressed in analytical forms within a certain range of nnn approximation parameter (m). The critical values of m for {X,Y}-cyclacenes with varying X (=C, N, B) and Y (=C, N, B) are calculated. For {X,X}-cyclacenes with a π-electron on each atom, all HMO quantities except total π-electron energies for a given value of m are found to be independent of X. The cyclic dimer (CD) is constructed in obtaining the eigenvalues corresponding to the singular points of the density of states (DOS) of such {X,Y}-cyclacene. Although the HOMO–LUMO gap of the CD differs from that of the cyclacene with a large number of repeating units (i.e. n ⟶ ∞) but becomes the same for m = 0. The analytical expressions can be used for facile computer programming in obtaining the HMO quantities. Such nnn interaction approximations actually release, to some extent, the strain that results in due to the geometrical structures of such cyclacenes, which is evident from the plots of strain energy per segment vs. contribution of such interactions on the total π-electron energy, where the slopes decrease with an increase in m. The vertical absorption energy difference for singlet-triplet states bears excellent linear correlation with the HOMO–LUMO gaps for a certain m value (m = 0.3) in the case of an even n, but for an odd n, such energy difference remains invariant.

Computing First General Zagreb Index of Operations on Graphs

The numerical coding of the molecular structures on the bases of topological indices plays an important role in the subject of Cheminformatics which is a combination of Computer, Chemistry, and Information Science. In 1972, it was shown that the total π-electron energy of a molecular graph depends upon its structure and it can be obtained by the sum of the square of degrees of the vertices of a molecular graph. Later on, this sum was named as the first Zagreb index. In 2005, for γeR - {0, 1}, the first general Zagreb index of a graph G is defined as M γ (G) = Σ veV(G) [d G (v)] γ , where d G (v) is degree of the vertex v in G. In this paper, for each γeR - {0, 1}, we study the first general Zagreb index of the cartesian product of two graphs such that one of the graphs is D-sum graph and the other is any connected graph, where D-sum graph is obtained by using certain D operations on a connected graph. The obtained results are general extensions of the results of Deng et al. [Applied Mathematics and Computation 275(2016): 422-431] and Akhter et al. [AKCE Int. J. Graphs Combin. 14(2017): 70-79] who proved only for γ = 2 and γ = 3, respectively.

Zagreb Connection Indices of Some Nanostructures

Abstract The first Zagreb index (occurred in an approximate formula of total π-electron energy, communicated in 1972) and the second Zagreb index (appeared in 1975, within the study of molecular branching) are among the most studied topological indices. Recently, three modified versions of the Zagreb indices were proposed independently in [A. Ali, N. Trinajstić, A novel/old modification of the first Zagreb index, arXiv:1705.10430 [math.CO], 2017] and [A. M. Naji, N. D. Soner, I. Gutman, On leap Zagreb indices of graphs, Commun. Comb. Optim., 2017, 2, 99–117], which were named as the Zagreb connection indices and the leap Zagreb indices, respectively. In this paper, we derive formulas for calculating these modified versions of the Zagreb indices of four well known nanostructures.

Extended energy and its dependence on molecular structure

The extended energy ([Formula: see text]) is a vertex degree based and spectrum-based molecular structure descriptor, shown to be well correlated with a variety of physicochemical molecular properties. We investigate the dependence of [Formula: see text] on molecular structure and establish its basic characteristics. In particular, we show how [Formula: see text] is related with the geometric–arithmetic (GA) topological index. Our main finding is that the difference between [Formula: see text] and the total π-electron energy is linearly proportional to the difference between the number of edges and the GA index.