Molecular trees with extremal values of Sombor indices

2021 ◽  
Author(s):  
Hanyuan Deng ◽  
Zikai Tang ◽  
Renfang Wu
Keyword(s):  
Extremal Values ◽  
Molecular Trees

Jahn-Teller Effect in Hexahydroxocuprate(II) Complexes

1995 ◽  
Vol 60 (9) ◽  
pp. 1429-1434
Author(s):  
Martin Breza
Keyword(s):  
Hydrogen Bonding ◽  
Quantum Chemical ◽  
Potential Surface ◽  
Hydroxyl Group ◽  
Chemical Calculation ◽  
Teller Distortion ◽  
Jahn Teller ◽  
Jahn Teller Effect ◽  
Extremal Values ◽  
Jahn Teller Distortion

Using semiempirical CNDO-UHF method the adiabatic potential surface of 2[Cu(OH)6]4- complexes is investigated. The values of vibration and vibronic constants for Eg - (a1g + eg) vibronic interaction attain extremal values for the optimal O-H distance. The Jahn-Teller distortion decreases with increasing O-H distance. The discrepancy between experimentally observed elongated bipyramid of [Cu(OH)6]4- in Ba2[Cu(OH)6] and the compressed one obtained by quantum-chemical calculation is explainable by hydrogen bonding of the axial hydroxyl group.

scholarly journals Extremal Values of the Sackin Tree Balance Index

2021 ◽  
Author(s):  
Mareike Fischer
Keyword(s):  
Phylogenetic Trees ◽  
Theoretical Computer Science ◽  
Search Trees ◽  
Formal Proofs ◽  
Birth Process ◽  
Ordered Trees ◽  
Research Areas ◽  
New Findings ◽  
Mathematical Phylogenetics ◽  
Extremal Values

AbstractTree balance plays an important role in different research areas like theoretical computer science and mathematical phylogenetics. For example, it has long been known that under the Yule model, a pure birth process, imbalanced trees are more likely than balanced ones. Also, concerning ordered search trees, more balanced ones allow for more efficient data structuring than imbalanced ones. Therefore, different methods to measure the balance of trees were introduced. The Sackin index is one of the most frequently used measures for this purpose. In many contexts, statements about the minimal and maximal values of this index have been discussed, but formal proofs have only been provided for some of them, and only in the context of ordered binary (search) trees, not for general rooted trees. Moreover, while the number of trees with maximal Sackin index as well as the number of trees with minimal Sackin index when the number of leaves is a power of 2 are relatively easy to understand, the number of trees with minimal Sackin index for all other numbers of leaves has been completely unknown. In this manuscript, we extend the findings on trees with minimal and maximal Sackin indices from the literature on ordered trees and subsequently use our results to provide formulas to explicitly calculate the numbers of such trees. We also extend previous studies by analyzing the case when the underlying trees need not be binary. Finally, we use our results to contribute both to the phylogenetic as well as the computer scientific literature using the new findings on Sackin minimal and maximal trees to derive formulas to calculate the number of both minimal and maximal phylogenetic trees as well as minimal and maximal ordered trees both in the binary and non-binary settings. All our results have been implemented in the Mathematica package SackinMinimizer, which has been made publicly available.

scholarly journals The General (α,3)-Path Connectivity Indices of Polycyclic Aromatic Hydrocarbons

2018 ◽  
Vol 2018 ◽  
pp. 1-5
Author(s):  
Haiying Wang ◽  
Chuantao Li
Keyword(s):  
Applied Sciences ◽  
Polycyclic Aromatic Hydrocarbons ◽  
Aromatic Hydrocarbons ◽  
Molecular Graph ◽  
Three Dimensional ◽  
3D Qsar ◽  
Medical Sciences ◽  
Connectivity Indices ◽  
Polycyclic Aromatic ◽  
Extremal Values

The general (α,t)-path connectivity index of a molecular graph originates from many practical problems such as three-dimensional quantitative structure-activity (3D QSAR) and molecular chirality. It is defined as Rtα(G)=∑Pt=vi1vi2⋯vit+1⊆G[d(vi1)d(vi2)⋯d(vit+1)]α, where the summation is taken over all possible paths of length t of G and we do not distinguish between the paths vi1vi2⋯vit+1 and vit+1⋯vi2vi1. In this paper, we focus on the structures of Polycyclic Aromatic Hydrocarbons (PAHn), which play a role in organic materials and medical sciences. We try to compute the exact general (α,3)-path connectivity indices of this family of hydrocarbon structures. Furthermore, we exactly derive the monotonicity and the extremal values of R3α(PAHn) for any real number α. These valuable results could produce strong guiding significance to these applied sciences.

scholarly journals On Mostar and Edge Mostar Indices of Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Ali Ghalavand ◽  
Ali Reza Ashrafi ◽  
Mardjan Hakimi-Nezhaad
Keyword(s):  
Upper Bound ◽  
Graph Invariants ◽  
Open Questions ◽  
Bicyclic Graphs ◽  
Extremal Values

Let G be a graph with edge set E G and e = u v ∈ E G . Define n u e , G and m u e , G to be the number of vertices of G closer to u than to v and the number of edges of G closer to u than to v , respectively. The numbers n v e , G and m v e , G can be defined in an analogous way. The Mostar and edge Mostar indices of G are new graph invariants defined as M o G = ∑ u v ∈ E G n u u v , G − n v u v , G and M o e G = ∑ u v ∈ E G m u u v , G − m v u v , G , respectively. In this paper, an upper bound for the Mostar and edge Mostar indices of a tree in terms of its diameter is given. Next, the trees with the smallest and the largest Mostar and edge Mostar indices are also given. Finally, a recent conjecture of Liu, Song, Xiao, and Tang (2020) on bicyclic graphs with a given order, for which extremal values of the edge Mostar index are attained, will be proved. In addition, some new open questions are presented.

Molecular trees

Nature ◽  
1977 ◽  
Vol 270 (5637) ◽  
pp. 547-547
Author(s):  
Thomas H. Jukes
Keyword(s):  
Molecular Trees

Electron-Reservoir Sandwich Complexes: From Mono- and Bimetallic Complexes to Molecular Trees

1996 ◽  
pp. 107-127 ◽  
Author(s):  
D. Astruc ◽  
C. Valério ◽  
J.-L. Fillaut ◽  
J. Ruiz ◽  
J.-R. Hamon ◽  
...  
Keyword(s):  
Bimetallic Complexes ◽  
Sandwich Complexes ◽  
Molecular Trees
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