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 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.

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 Topology and stability of conjugated hydrocarbons: The dependence of total π-electron energy on molecular topology

2005 ◽  
Vol 70 (3) ◽  
pp. 441-456 ◽  
Author(s):  
Ivan Gutman
Keyword(s):  
Present Article ◽  
Molecular Orbital ◽  
Electron Energy ◽  
Molecular Topology ◽  
Total Electron ◽  
Conjugated Hydrocarbons ◽  
Mathematical Properties ◽  
Total Electron Energy ◽  
Orbital Approximation

In spite of the fact that research on the mathematical properties of the total ?-electron energy E (as computed by means of the H?ckel molecular orbital approximation) started already in the 1940s, many results in this area have been obtained also in the newest times. In 1978 this author published in this journal a review on E. The present article is another review on E, summarizing the progress in the theory of E, achieved since then.

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

2007 ◽  
Vol 72 (12) ◽  
pp. 1343-1350 ◽  
Author(s):  
Slavko Radenkovic ◽  
Ivan Gutman
Keyword(s):  
Electron Energy ◽  
Linear Correlation ◽  
Molecular Graph ◽  
Total Electron ◽  
Good Linear ◽  
Molecular Graphs ◽  
Acyclic Graphs ◽  
Laplacian Energy ◽  
Total Electron Energy ◽  
Good Linear Correlation

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.

scholarly journals Dependence of the total -electron energy on large number of non-bonding molecular orbitals

2004 ◽  
Vol 69 (10) ◽  
pp. 777-782 ◽  
Author(s):  
Ivan Gutman ◽  
Dragan Stevanovic ◽  
Slavko Radenkovic ◽  
Svetlana Milosavljevic ◽  
Natasa Cmiljanovic
Keyword(s):  
Electron Energy ◽  
Molecular Orbitals ◽  
Total Electron ◽  
Total Electron Energy

Using a recently developed method for computing the effect of non-bonding molecular orbitals (NBMOs) on the total ?-electron energy (E), it was found that the dependence of E on the number n0 of NBMOs is almost perfectly linear. We now show that this regularity remains valid for very large values of n0, in particular, to hold up to n0 = 20.

Conflict-Free Colourings of Uniform Hypergraphs With Few Edges

2012 ◽  
Vol 21 (4) ◽  
pp. 611-622 ◽  
Author(s):  
A. KOSTOCHKA ◽  
M. KUMBHAT ◽  
T. ŁUCZAK
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Lower And Upper Bounds ◽  
Uniform Hypergraphs ◽  
Minimum Number

A colouring of the vertices of a hypergraph is called conflict-free if each edge e of contains a vertex whose colour does not repeat in e. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of , and is denoted by χCF(). Pach and Tardos proved that for an (2r − 1)-uniform hypergraph with m edges, χCF() is at most of the order of rm1/r log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for r-uniform hypergraphs. In this paper we show that this is not necessarily the case. Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colourable.

scholarly journals A Computational Perspective on Network Coding

2010 ◽  
Vol 2010 ◽  
pp. 1-11
Author(s):  
Qin Guo ◽  
Mingxing Luo ◽  
Lixiang Li ◽  
Yixian Yang
Keyword(s):  
Graph Theory ◽  
Computational Complexity ◽  
Lower Bound ◽  
Network Coding ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Multicast Networks ◽  
Computational Perspective ◽  
Source Rate

From the perspectives of graph theory and combinatorics theory we obtain some new upper bounds on the number of encoding nodes, which can characterize the coding complexity of the network coding, both in feasible acyclic and cyclic multicast networks. In contrast to previous work, during our analysis we first investigate the simple multicast network with source rateh=2, and thenh≥2. We find that for feasible acyclic multicast networks our upper bound is exactly the lower bound given by M. Langberg et al. in 2006. So the gap between their lower and upper bounds for feasible acyclic multicast networks does not exist. Based on the new upper bound, we improve the computational complexity given by M. Langberg et al. in 2009. Moreover, these results further support the feasibility of signatures for network coding.

Minimization of Bounded Cable Tensions in Cable-Based Parallel Manipulators

2007 ◽  
Author(s):  
Mahir Hassan ◽  
Amir Khajepour
Keyword(s):  
Lower Bound ◽  
Numerical Method ◽  
Upper Bound ◽  
Task Force ◽  
Parallel Manipulators ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Alternating Projection ◽  
Alternating Projection Method ◽  
Cable Forces

In this work, the application of the Dykstra’s alternating projection method to find the minimum-2-norm solution for actuator forces is discussed in the case when lower and upper bounds are imposed on the actuator forces. The lower bound is due to specified pretension desired in the cables and the upper bound is due to the maximum allowable forces in the cables. This algorithm presents a systematic numerical method to determine whether or not a solution exists to the cable forces within these bounds and, if it does exist, calculate the minimum-2-norm solution for the cable forces for a given task force. This method is applied to an example 2-DOF translational cable-driven manipulator and a geometrical demonstration is presented.

Sign in / Sign up

Export Citation Format

Share Document