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.

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.

Laplacian Energy of Digraphs and a Minimum Laplacian Energy Algorithm

2015 ◽  
Vol 26 (03) ◽  
pp. 367-380 ◽  
Author(s):  
Xingqin Qi ◽  
Edgar Fuller ◽  
Rong Luo ◽  
Guodong Guo ◽  
Cunquan Zhang
Keyword(s):  
Oriented Graph ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Spectral Graph Theory ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Spectral Graph ◽  
Laplacian Energy ◽  
Energy Formula ◽  
Matrix Formula

In spectral graph theory, the Laplacian energy of undirected graphs has been studied extensively. However, there has been little work yet for digraphs. Recently, Perera and Mizoguchi (2010) introduced the directed Laplacian matrix [Formula: see text] and directed Laplacian energy [Formula: see text] using the second spectral moment of [Formula: see text] for a digraph [Formula: see text] with [Formula: see text] vertices, where [Formula: see text] is the diagonal out-degree matrix, and [Formula: see text] with [Formula: see text] whenever there is an arc [Formula: see text] from the vertex [Formula: see text] to the vertex [Formula: see text] and 0 otherwise. They studied the directed Laplacian energies of two special families of digraphs (simple digraphs and symmetric digraphs). In this paper, we extend the study of Laplacian energy for digraphs which allow both simple and symmetric arcs. We present lower and upper bounds for the Laplacian energy for such digraphs and also characterize the extremal graphs that attain the lower and upper bounds. We also present a polynomial algorithm to find an optimal orientation of a simple undirected graph such that the resulting oriented graph has the minimum Laplacian energy among all orientations. This solves an open problem proposed by Perera and Mizoguchi at 2010.

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.

scholarly journals Estimating vertex-degree-based energies

2022 ◽  
Vol 70 (1) ◽  
pp. 13-23
Author(s):  
Ivan Gutman
Keyword(s):  
General Theory ◽  
Spectral Theory ◽  
Current Literature ◽  
Bipartite Graphs ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Lower And Upper Bounds ◽  
Graph Invariants ◽  
General Properties ◽  
Graph Energy

Introduction/purpose: In the current literature, several dozens of vertex-degree-based (VDB) graph invariants are being studied. To each such invariant, a matrix can be associated. The VDB energy is the energy (= sum of the absolute values of the eigenvalues) of the respective VDB matrix. The paper examines some general properties of the VDB energy of bipartite graphs. Results: Estimates (lower and upper bounds) are established for the VDB energy of bipartite graphs in which there are no cycles of size divisible by 4, in terms of ordinary graph energy. Conclusion: The results of the paper contribute to the spectral theory of VDB matrices, especially to the general theory of VDB energy.

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.

scholarly journals New Bounds for Distance Energy of A Graph

2020 ◽  
Vol 26 (2) ◽  
pp. 213-223
Author(s):  
G. Sridhara ◽  
Rajesh Kanna ◽  
H.L. Parashivamurthy
Keyword(s):  
Connected Graph ◽  
Molecular Descriptor ◽  
Chemical Properties ◽  
Distance Matrix ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Physico Chemical ◽  
The Absolute

For any connected graph G, the distance energy, E_D(G) is defined as the sum of the absolute eigenvalues of its distance matrix.  Distance energy was introduced by Indulal et al in the year 2008. It has significant importance in QSPR analysis of molecular descriptor to study  their physico-chemical properties. Our interest in this article is to establish new lower and upper bounds for distance energy.

scholarly journals Energy of Pythagorean Fuzzy Graphs with Applications

Mathematics ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 136 ◽  
Author(s):  
Muhammad Akram ◽  
Sumera Naz
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Satellite Communication ◽  
Research Study ◽  
Upper Bounds ◽  
Intuitionistic Fuzzy Sets ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs ◽  
Laplacian Energy ◽  
Duality Property

Pythagorean fuzzy sets (PFSs), an extension of intuitionistic fuzzy sets (IFSs), inherit the duality property of IFSs and have a more powerful ability than IFSs to model the obscurity in practical decision-making problems. In this research study, we compute the energy and Laplacian energy of Pythagorean fuzzy graphs (PFGs) and Pythagorean fuzzy digraphs (PFDGs). Moreover, we derive the lower and upper bounds for the energy and Laplacian energy of PFGs. Finally, we present numerical examples, including the design of a satellite communication system and the evaluation of the schemes of reservoir operation to illustrate the applications of our proposed concepts in decision making.

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 The Third-Order Hermitian Toeplitz Determinant for Alpha-Convex Functions

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1274
Author(s):  
Anna Dobosz
Keyword(s):  
Convex Functions ◽  
Upper Bounds ◽  
Arithmetic Mean ◽  
Lower And Upper Bounds ◽  
Third Order ◽  
Toeplitz Determinants ◽  
The Third ◽  
Toeplitz Determinant ◽  
Symmetry Properties ◽  
Definition Of

Sharp lower and upper bounds of the second- and third-order Hermitian Toeplitz determinants for the class of α-convex functions were found. The symmetry properties of the arithmetic mean underlying the definition of α-convexity and the symmetry properties of Hermitian matrices were used.

scholarly journals Properties of a Generalized Class of Weights Satisfying Reverse Hölder’s Inequality

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
S. H. Saker ◽  
S. S. Rabie ◽  
R. P. Agarwal
Keyword(s):  
Upper Bounds ◽  
Hölder Inequality ◽  
The Self ◽  
Power Mean ◽  
Lower And Upper Bounds ◽  
Real Numbers ◽  
Positive Real ◽  
Fundamental Properties ◽  
Reverse Hölder ◽  
Special Case

In this paper, we will prove some fundamental properties of the discrete power mean operator M p u n = 1 / n ∑ k = 1 n   u p k 1 / p , for   n ∈ I ⊆ ℤ + , of order p , where u is a nonnegative discrete weight defined on I ⊆ ℤ + the set of the nonnegative integers. We also establish some lower and upper bounds of the composition of different operators with different powers. Next, we will study the structure of the generalized discrete class B p q B of weights that satisfy the reverse Hölder inequality   M q u ≤ B M p u , for positive real numbers p , q , and B such that 0 < p < q and B > 1 . For applications, we will prove some self-improving properties of weights from B p q B and derive the self improving properties of the discrete Gehring weights as a special case. The paper ends by a conjecture with an illustrative sharp example.

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