Extremal trees with respect to the Steiner Wiener index

2019 ◽  
Vol 11 (06) ◽  
pp. 1950067
Author(s):  
Jie Zhang ◽  
Guang-Jun Zhang ◽  
Hua Wang ◽  
Xiao-Dong Zhang
Keyword(s):  
Maximum Degree ◽  
Computational Analysis ◽  
Degree Sequence ◽  
Wiener Index ◽  
Extremal Problems ◽  
Difficult Problem ◽  
Computational Results ◽  
Number Of Leaves ◽  
Extremal Structure ◽  
Chemical Trees

The well-known Wiener index is defined as the sum of pairwise distances between vertices. Extremal problems with respect to it have been extensively studied for trees. A generalization of the Wiener index, called the Steiner Wiener index, takes the sum of minimum sizes of subgraphs that span [Formula: see text] given vertices over all possible choices of the [Formula: see text] vertices. We consider the extremal problems with respect to the Steiner Wiener index among trees of a given degree sequence. First, it is pointed out minimizing the Steiner Wiener index in general may be a difficult problem, although the extremal structure may very likely be the same as that for the regular Wiener index. We then consider the upper bound of the general Steiner Wiener index among trees of a given degree sequence and study the corresponding extremal trees. With these findings, some further discussion and computational analysis are presented for chemical trees. We also propose a conjecture based on the computational results. In addition, we identify the extremal trees that maximize the Steiner Wiener index among trees with a given maximum degree or number of leaves.

Computational Simulations of Dynamic Compaction of Dry Sand

2012 ◽  
Vol 598 ◽  
pp. 516-519
Author(s):  
Yu Qing Ding ◽  
Wen Hui Tang ◽  
Xian Wen Ran ◽  
Xin Xu
Keyword(s):  
Computational Analysis ◽  
Dynamic Compaction ◽  
Back Surface ◽  
Computational Results ◽  
Two Dimensional ◽  
Computational Simulations ◽  
Plate Impact ◽  
Arrival Times ◽  
Dry Sand ◽  
Impact Experiments

The computational analysis of plate impact experiments on dry sand utilizing the Mie- Grüneisen (MG) equation of state and the P-α compaction model were investigated in this study. A number of two dimensional axial symmetric computations were performed by using the hydrocode AUTODYN. The computational results were compared with the particle velocity on the back surface of the rear plate measured by the VISAR system and the first shock-wave arrival times detected by piezoelectric pins in the samples respectively. It was found that the P-α compaction model was more accurately reproduce the experimental data than the MG EOS.

A Linear Algorithm for the Hyper-Wiener Index of Chemical Trees

2001 ◽  
Vol 41 (4) ◽  
pp. 958-963 ◽  
Author(s):  
Roberto Aringhieri ◽  
Pierre Hansen ◽  
Federico Malucelli
Keyword(s):  
Wiener Index ◽  
Linear Algorithm ◽  
Chemical Trees

A Comparative Study of Constrained and Unconstrained Melting Inside a Sphere

2020 ◽  
Author(s):  
Rohit Kothari ◽  
Shripad T. Revankar ◽  
Santosh K. Sahu ◽  
Shailesh I. Kundalwal
Keyword(s):  
Boundary Conditions ◽  
Comparative Study ◽  
Computational Analysis ◽  
Computational Results ◽  
Melting Fraction ◽  
Initial And Boundary Conditions ◽  
Spherical Capsule ◽  
Good Agreement

Abstract Present study is focused on the computational analysis of melting of PCM inside the spherical capsule. Both unconstrained and constrained melting is analyzed for the constant PCM volume and similar initial and boundary conditions. RT27 is chosen as the PCM for this study. Air is considered at the top of PCM inside the spherical capsule. Results are validated with the existing experimental and computational results and found to be in good agreement. Results obtained from present study are compared for the melting fraction, pattern and time. Composite diagrams are presented for the streamline and temperature contours.

scholarly journals Extremal Chemical Trees

2002 ◽  
Vol 57 (1-2) ◽  
pp. 49-51
Author(s):  
Miranca Fischermann ◽  
Ivan Gutman ◽  
Arne Hoffmann ◽  
Dieter Rautenbach ◽  
Dušica Vidovića ◽  
...  
Keyword(s):  
Carbon Atom ◽  
Molecular Graph ◽  
Chemical Properties ◽  
Wiener Index ◽  
Graph Representation ◽  
Hosoya Index ◽  
Saturated Hydrocarbons ◽  
Physico Chemical ◽  
Graph Energy ◽  
Chemical Trees

A variety of molecular-graph-based structure-descriptors were proposed, in particular the Wiener index W. the largest graph eigenvalue λ1, the connectivity index X, the graph energy E and the Hosoya index Z, capable of measuring the branching of the carbon-atom skeleton of organic compounds, and therefore suitable for describing several of their physico-chemical properties. We now determine the structure of the chemical trees (= the graph representation of acyclic saturated hydrocarbons) that are extremal with respect to W , λ1, E, and Z. whereas the analogous problem for X was solved earlier. Among chemical trees with 5. 6, 7, and 3k + 2 vertices, k = 2,3,..., one and the same tree has maximum λ1 and minimum W, E, Z. Among chemical trees with 3k and 3k +1 vertices, k = 3,4...., one tree has minimum 11 and maximum λ1 and another minimum E and Z .

scholarly journals The Gn,m Phase Transition is Not Hard for the Hamiltonian Cycle Problem

1998 ◽  
Vol 9 ◽  
pp. 219-245 ◽  
Author(s):  
B. Vandegriend ◽  
J. Culberson
Keyword(s):  
Phase Transition ◽  
High Frequency ◽  
Maximum Degree ◽  
Hamiltonian Cycle ◽  
Degree Sequence ◽  
Average Degree ◽  
Regular Graphs ◽  
M Phase ◽  
Backtrack Algorithm ◽  
Pruning Techniques

Using an improved backtrack algorithm with sophisticated pruning techniques, we revise previous observations correlating a high frequency of hard to solve Hamiltonian Cycle instances with the Gn,m phase transition between Hamiltonicity and non-Hamiltonicity. Instead all tested graphs of 100 to 1500 vertices are easily solved. When we artificially restrict the degree sequence with a bounded maximum degree, although there is some increase in difficulty, the frequency of hard graphs is still low. When we consider more regular graphs based on a generalization of knight's tours, we observe frequent instances of really hard graphs, but on these the average degree is bounded by a constant. We design a set of graphs with a feature our algorithm is unable to detect and so are very hard for our algorithm, but in these we can vary the average degree from O(1) to O(n). We have so far found no class of graphs correlated with the Gn,m phase transition which asymptotically produces a high frequency of hard instances.

scholarly journals On Different "Middle Parts" of a Tree

10.37236/6408 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Heather Smith ◽  
László Székely ◽  
Hua Wang ◽  
Shuai Yuan
Keyword(s):  
Maximum Degree ◽  
Rooted Tree ◽  
Maximum Distance ◽  
Partial Characterization ◽  
Extremal Structure

We determine the maximum distance between any two of the center, centroid, and subtree core among trees with a given order. Corresponding results are obtained for trees with given maximum degree and also for trees with given diameter. The problem of the maximum distance between the centroid and the subtree core among trees with given order and diameter becomes difficult. It can be solved in terms of the problem of minimizing the number of root-containing subtrees in a rooted tree of given order and height. While the latter problem remains unsolved, we provide a partial characterization of the extremal structure.

scholarly journals Solution Strategies for a Multiport Container Ship Stowage Problem

2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
Consuelo Parreño-Torres ◽  
Ramon Alvarez-Valdes ◽  
Francisco Parreño
Keyword(s):  
Integer Programming ◽  
Computational Analysis ◽  
Programming Model ◽  
Programming Models ◽  
Container Ship ◽  
Computational Results ◽  
Solution Strategies ◽  
Large Size ◽  
Loading And Unloading ◽  
First Time

The multiport container ship stowage problem consists in determining the position of the containers on board a ship along its route with the objective of minimizing the number of unproductive moves required in the loading and unloading operations at each port. This paper presents an integer programming model for the problem and proposes several sets of valid constraints that bring its LP-relaxation closer to an integer solution. Moreover, it presents a GRASP algorithm that generates stowage plans with a minimal number of unproductive moves in a high percentage of medium and large-size instances. An extended computational analysis has been performed in which, to the best of the authors’ knowledge, the efficiency of integer programming models for the problem is tested for the first time. With respect to GRASP, the computational results show that it performs well on different sized datasets.

Extremal Chemical Trees

2002 ◽  
Vol 57 (9-10) ◽  
pp. 49-52 ◽  
Author(s):  
Miranca Fischermann ◽  
Ivan Gutmana ◽  
Arne Hoffmann ◽  
Dieter Rautenbach ◽  
Dušica Vidović ◽  
...  
Keyword(s):  
Carbon Atom ◽  
Molecular Graph ◽  
Chemical Properties ◽  
Wiener Index ◽  
Graph Representation ◽  
Hosoya Index ◽  
Saturated Hydrocarbons ◽  
Physico Chemical ◽  
Graph Energy ◽  
Chemical Trees

Avariety of molecular-graph-based structure-descriptors were proposed, in particular the Wiener index W, the largest graph eigenvalue λ1, the connectivity index χ, the graph energy E and the Hosoya index Z, capable of measuring the branching of the carbon-atom skeleton of organic compounds, and therefore suitable for describing several of their physico-chemical properties. We now determine the structure of the chemical trees (= the graph representation of acyclic saturated hydrocarbons) that are extremal with respect to W, λ1, E, and Z, whereas the analogous problem for χ was solved earlier. Among chemical trees with 5, 6, 7, and 3k + 2 vertices, k = 2, 3,..., one and the same tree has maximum λ1 and minimum W, E, Z. Among chemical trees with 3k and 3k + 1 vertices, k = 3, 4..., one tree has minimum W and maximum λ1 and another minimum E and Z.

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