scholarly journals Estimating and Approximating the Total π-Electron Energy of Benzenoid Hydrocarbons

2000 ◽  
Vol 55 (5) ◽  
pp. 507-512 ◽  
Author(s):  
I. Gutman ◽  
J. H. Koolen ◽  
V. Moulto ◽  
M. Parac ◽  
T. Soldatović ◽  
...  
Keyword(s):  
Electron Energy ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Benzenoid Hydrocarbons ◽  
Carbon Carbon Bonds ◽  
Carbon Bonds ◽  
Approximate Formulas ◽  
Better Than

Abstract Lower and upper bounds as well as approximate formulas for the total π-electron energy (E) of benzenoid hydrocarbons are deduced, depending only on the number of carbon atoms (n) and number of carbon-carbon bonds (to). These are better than the several previously known (n, m)-type estimates and approximations for E.

Upper Bounds for the Total π-Electron Energy of Benzenoid Hydrocarbons and Their Relations

1986 ◽  
Vol 41 (6) ◽  
pp. 861-865
Author(s):  
J. Ciosłowski ◽  
I. Gutman
Keyword(s):  
Electron Energy ◽  
Upper Bounds ◽  
Benzenoid Hydrocarbons ◽  
Carbon Carbon Bonds ◽  
Carbon Bonds

Upper bounds for the total π-electron energy of benzenoid hydrocarbons, depending on the number of carbon atoms and carbon-carbon bonds, are examined and several of their properties established.

scholarly journals Iterative estimation of total π-electron energy

2005 ◽  
Vol 70 (10) ◽  
pp. 1193-1197 ◽  
Author(s):  
Lemi Türker ◽  
Ivan Gutman
Keyword(s):  
Lower Bound ◽  
Electron Energy ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Benzenoid Hydrocarbons ◽  
Total Electron ◽  
Iterative Estimation ◽  
Total Electron Energy ◽  
Almost All

In this work, the lower and upper bounds for total ?-electron energy (E) was studied. A method is presented, by means of which, starting with a lower bound EL and an upper bound EU for E, a sequence of auxiliary quantities E0 E1, E2,? is computed, such that E0 = EL, E0 < E1 < E2 < ?, and E = EU. Therefore, an integer k exists, such that Ek E < Ek+1. If the estimates EL and EU are of the McClelland type, then k is called the McClelland number. For almost all benzenoid hydrocarbons, k = 3.

Relation between Pauling and Coulson Bond Orders in Benzenoid Hydrocarbons

2004 ◽  
Vol 59 (10) ◽  
pp. 699-704
Author(s):  
Ivan Gutman ◽  
Boris Furtula ◽  
Slavko Radenković
Keyword(s):  
Electron Energy ◽  
Approximate Expression ◽  
Bond Orders ◽  
Hydrogen Atoms ◽  
Benzenoid Hydrocarbons ◽  
Carbon Carbon Bonds ◽  
Carbon Bonds

The relation between Pauling and Coulson bond orders in benzenoid hydrocarbons is examined. The carbon-carbon bonds of benzenoid hydrocarbons have to be classified into three classes, depending on the number of attached hydrogen atoms. Within each class the correlation between the bond orders is linear. The results can be used to rationalize the recently discovered correlation between the energy and electron contents of rings. An approximate expression for the total π-electron energy is also deduced.

Optimization Design of Electromagnetic Devices Using an Enhanced Salp Swarm Algorithm

2021 ◽  
Vol 35 (12) ◽  
pp. 1471-1476
Author(s):  
Houssem Bouchekara ◽  
Mostafa Smail ◽  
Mohamed Javaid ◽  
Sami Shamsah
Keyword(s):  
Optimization Design ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Design Problems ◽  
Salp Swarm Algorithm ◽  
Electromagnetic Devices ◽  
Swarm Algorithm ◽  
Better Than

An Enhanced version of the Salp Swarm Algorithm (SSA) referred to as (ESSA) is proposed in this paper for the optimization design of electromagnetic devices. The ESSA has the same structure as of the SSA with some modifications in order to enhance its performance for the optimization design of EMDs. In the ESSA, the leader salp does not move around the best position with a fraction of the distance between the lower and upper bounds as in the SAA; rather, a modified mechanism is used. The performance of the proposed algorithm is tested on the widely used Loney’s solenoid and TEAM Workshop Problem 22 design problems. The obtained results show that the proposed algorithm is much better than the initial one. Furthermore, a comparison with other well-known algorithms revealed that the proposed algorithm is very competitive for the optimization design of electromagnetic devices.

Generalizing the McClelland Bounds for Total π-Electron Energy

2008 ◽  
Vol 63 (5-6) ◽  
pp. 280-282 ◽  
Author(s):  
Ivan Gutman ◽  
Gopalapillai Indulal ◽  
Roberto Todeschinic
Keyword(s):  
Electron Energy ◽  
Distance Matrix ◽  
Upper Bounds ◽  
Arithmetic Mean ◽  
Lower And Upper Bounds ◽  
Real Numbers ◽  
Laplacian Energy ◽  
Graph 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 = Σni =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.

Lower Bounds for the Probability of Intersection of Several Unions of Events

1998 ◽  
Vol 7 (4) ◽  
pp. 353-364 ◽  
Author(s):  
TUHAO CHEN ◽  
E. SENETA
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Numerical Example ◽  
Bonferroni Inequalities ◽  
Indicator Functions ◽  
Single Set ◽  
Better Than

To bound the probability of a union of n events from a single set of events, Bonferroni inequalities are sometimes used. There are sharper bounds which are called Sobel–Uppuluri–Galambos inequalities. When two (or more) sets of events are involved, bounds are considered on the probability of intersection of several such unions, one union from each set. We present a method for unified treatment of bivariate lower and upper bounds in this note. The lower bounds obtained are new and at least as good as lower bounds appearing in the literature so far. The upper bounds coincide with existing bivariate Sobel–Uppuluri–Galambos type upper bounds derived by the method of indicator functions. A numerical example is given to illustrate that the new lower bounds can be strictly better than existing ones.

Estrada Index of Benzenoid Hydrocarbons

2007 ◽  
Vol 62 (5-6) ◽  
pp. 254-258
Author(s):  
Ivan Gutman ◽  
Slavko Radenković
Keyword(s):  
Molecular Graph ◽  
Benzenoid Hydrocarbons ◽  
Estrada Index ◽  
Carbon Carbon Bonds ◽  
Carbon Bonds ◽  
Structure Descriptor

A structure-descriptor EE, recently proposed by Estrada, is examined. If λ1, λ2, . . . ,λn are the eigenvalues of the molecular graph, then . In the case of benzenoid hydrocarbons with n carbon atoms and m carbon-carbon bonds, EE is found to be accurately approximated by means of the formula a1 n cosh (√2m/n)+a2, where a1 ≈ 1.098 and a2 = −0.64 are empirically determined fitting constants. Within classes of benzenoid isomers (which all have equal n and m), the Estrada index is linearly proportional to the number of bay regions.

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

2016 ◽  
Vol 71 (2) ◽  
pp. 161-164 ◽  
Author(s):  
Ivan Gutman
Keyword(s):  
Electron Energy ◽  
Upper Bounds ◽  
Molecular Orbitals ◽  
Electron Systems ◽  
Lower And Upper Bounds ◽  
Conjugated Molecules

AbstractLower and upper bounds for the totalπ-electron energy are obtained, which are applicable to conjugatedπ-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.

scholarly journals Estimating the total π-electron energy

2013 ◽  
Vol 78 (12) ◽  
pp. 1925-1933 ◽  
Author(s):  
Ivan Gutman ◽  
Kinkar Das
Keyword(s):  
Electron Energy ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Total Electron ◽  
Molecular Graphs ◽  
Conjugated Hydrocarbons ◽  
Short Survey ◽  
Total Electron Energy ◽  
Graph 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.

scholarly journals On a class of approximate formulas for total ח-electron energy of benzenoid hydrocarbons

2001 ◽  
Vol 66 (2) ◽  
pp. 101-106 ◽  
Author(s):  
Ivan Gutman ◽  
Tanja Soldatovic
Keyword(s):  
Electron Energy ◽  
Benzenoid Hydrocarbons ◽  
Spectral Moments ◽  
Total Electron ◽  
Total Electron Energy ◽  
Approximate Formulas

The method for obtaining approximate formulas of the (n, m)-type for the total -electron energy of benzenoid hydrocarbons (communicated in J. Serb. Chem. Soc. 54 (1989) 189) is simplified and extended so as to include arbitrary spectral moments. The accuracy of the formulas thus obtained is very good and these need no additional fitting by means of empirically determined parameters.

Sign in / Sign up

Export Citation Format

Share Document