scholarly journals Extremal Values of Ratios: Distance Problems vs. Subtree Problems in Trees

10.37236/3020 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
László Székely ◽  
Hua Wang
Keyword(s):  
Negative Correlation ◽  
Wiener Index ◽  
Extremal Problems ◽  
Open Problems ◽  
Extremal Values ◽  
Appl Math

The authors discovered a dual behaviour of two tree indices, the Wiener index and the number of subtrees, for a number of extremal problems  [Discrete Appl. Math. 155 (3) 2006, 374-385;  Adv. Appl. Math. 34 (2005), 138-155]. Barefoot, Entringer and Székely [Discrete Appl. Math. 80 (1997), 37-56] determined extremal values of $\sigma_T(w)/\sigma_T(u)$,   $\sigma_T(w)/\sigma_T(v)$, $\sigma(T)/\sigma_T(v)$, and $\sigma(T)/\sigma_T(w)$, where $T$ is a tree on $n$ vertices, $v$ is in the centroid of the tree $T$, and $u,w$ are leaves in $T$.In this paper  we test how far the negative correlation between distances and subtrees go if we look for the   extremal values of $F_T(w)/F_T(u)$, $F_T(w)/F_T(v)$, $F(T)/F_T(v)$, and $F(T)/F_T(w)$, where $T$ is a tree on $n$ vertices, $v$ is in the subtree core of the tree $T$, and $u,w$ are leaves in $T$-the complete analogue of  [Discrete Appl. Math. 80 (1997), 37-56], changing distances to the number of subtrees.  We include a number of open problems, shifting the interest towards the number of subtrees in graphs.

The second-minimum Wiener index of cacti with given cycles

2020 ◽  
pp. 2150039
Author(s):  
Hanyuan Deng ◽  
G. C. Keerthi Vasan ◽  
S. Balachandran
Keyword(s):  
Connected Graph ◽  
Wiener Index ◽  
Index Formula ◽  
Extremal Graphs ◽  
Unique Graph ◽  
Sum Of Distances ◽  
Appl Math

The Wiener index [Formula: see text] of a connected graph [Formula: see text] is the sum of distances between all pairs of vertices of [Formula: see text]. A connected graph [Formula: see text] is said to be a cactus if each of its blocks is either a cycle or an edge. Let [Formula: see text] be the set of all [Formula: see text]-vertex cacti containing exactly [Formula: see text] cycles. Liu and Lu (2007) determined the unique graph in [Formula: see text] with the minimum Wiener index. Gutman, Li and Wei (2017) determined the unique graph in [Formula: see text] with maximum Wiener index. In this paper, we present the second-minimum Wiener index of graphs in [Formula: see text] and identify the corresponding extremal graphs, which solve partially the problem proposed by Gutman et al. [Cacti with [Formula: see text]-vertices and [Formula: see text] cycles having extremal Wiener index, Discrete Appl. Math. 232 (2017) 189–200] in 2017.

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.

De la Vallée Poussin inequality for impulsive differential equations

2021 ◽  
Vol 71 (4) ◽  
pp. 881-888
Author(s):  
Sibel Doğru Akgöl ◽  
Abdullah Özbekler
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Impulsive Differential Equations ◽  
Open Problems ◽  
Early Date ◽  
Impulse Effect ◽  
Appl Math ◽  
De La Vallée Poussin

Abstract The de la Vallée Poussin inequality is a handy tool for the investigation of disconjugacy, and hence, for the oscillation/nonoscillation of differential equations. The results in this paper are extensions of former those of Hartman and Wintner [Quart. Appl. Math. 13 (1955), 330–332] to the impulsive differential equations. Although the inequality first appeared in such an early date for ordinary differential equations, its improved version for differential equations under impulse effect never has been occurred in the literature. In the present study, first, we state and prove a de la Vallée Poussin inequality for impulsive differential equations, then we give some corollaries on disconjugacy. We also mention some open problems and finally, present some examples that support our findings.

On extremal bipartite graphs with given number of cut edges

2020 ◽  
Vol 12 (02) ◽  
pp. 2050015
Author(s):  
Hanlin Chen ◽  
Renfang Wu
Keyword(s):  
Topological Index ◽  
Wiener Index ◽  
Bipartite Graphs ◽  
Topological Indices ◽  
Extremal Graphs ◽  
Unified Approach ◽  
Harary Index ◽  
Eccentricity Index ◽  
Extremal Values

Let [Formula: see text] be a topological index of a graph. If [Formula: see text] (or [Formula: see text], respectively) for each edge [Formula: see text], then [Formula: see text] is monotonically decreasing (or increasing, respectively) with the addition of edges. In this paper, by a unified approach, we determine the extremal values of some monotonic topological indices, including the Wiener index, the hyper-Wiener index, the Harary index, the connective eccentricity index, the eccentricity distance sum, among all connected bipartite graphs with a given number of cut edges, and characterize the corresponding extremal graphs, respectively.

scholarly journals A Survey on Extremal Problems of Eigenvalues

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Ping Yan ◽  
Meirong Zhang
Keyword(s):  
Variational Method ◽  
Lebesgue Space ◽  
Upper Bounds ◽  
Extremal Problems ◽  
Lower And Upper Bounds ◽  
Open Problems ◽  
One Dimensional ◽  
Sturm Liouville ◽  
The One ◽  
Neumann Eigenvalues

Given an integrable potentialq∈L1([0,1],ℝ), the Dirichlet and the Neumann eigenvaluesλnD(q)andλnN(q)of the Sturm-Liouville operator with the potentialqare defined in an implicit way. In recent years, the authors and their collaborators have solved some basic extremal problems concerning these eigenvalues when theL1metric forqis given;∥q∥L1=r. Note that theL1spheres andL1balls are nonsmooth, noncompact domains of the Lebesgue space(L1([0,1],ℝ),∥·∥L1). To solve these extremal problems, we will reveal some deep results on the dependence of eigenvalues on potentials. Moreover, the variational method for the approximating extremal problems on the balls of the spacesLα([0,1],ℝ),1<α<∞will be used. Then theL1problems will be solved by passingα↓1. Corresponding extremal problems for eigenvalues of the one-dimensionalp-Laplacian with integrable potentials have also been solved. The results can yield optimal lower and upper bounds for these eigenvalues. This paper will review the most important ideas and techniques in solving these difficult and interesting extremal problems. Some open problems will also be imposed.

scholarly journals Colonization and Diversity of AM Fungi by Morphological Analysis on Medicinal Plants in Southeast China

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Mingyuan Wang ◽  
Pan Jiang
Keyword(s):  
Electrical Conductivity ◽  
Species Richness ◽  
Medicinal Plants ◽  
Negative Correlation ◽  
Arbuscular Mycorrhizal ◽  
Am Fungi ◽  
Wiener Index ◽  
Significant Negative Correlation ◽  
Spore Density ◽  
Southeast China

The arbuscular mycorrhizal (AM) fungal distributions in the rhizosphere of 20 medicinal plants species in Zhangzhou, southeast China, were studied. The results showed 66 species of 8 genera of AM fungi were identified, of which 38 belonged toGlomus, 12 toAcaulospora, 9 toScutellospora, 2 toGigaspora, 2 toFunneliformis, 1 toSeptoglomus, 1 toRhizophagus, and 1 toArchaeospora.Glomuswas the dominant genera andG. melanosporum,Acaulospora scrobiculata,G. etunicatum,Funneliformis mosseae, andG. rubiformewere the prevalent species. The highest colonization (100%) was recorded inDesmodium pulchellum(L.) Benth. while the lowest (8.0%) was inAcorus tatarinowiiSchott. The AM fungi spore density ranged from 270 to 2860 per 100 g soil (average 1005), and the species richness ranged from 3 to 14 (average 9.7) per soil sample. Shannon-Wiener index ranged from 0.52 to 2 (average 1.45). In the present study, the colonization had a highly negative correlation with availableKand electrical conductivity. Species richness correlated positively with electrical conductivity and organic matter. Shannon-Wiener index had a highly significant negative correlation with pH. This study provides a valuable germplasm and theoretical basis for AM fungal biotechnology on medicinal standardization planting.

scholarly journals On investigations of graphs preserving the Wiener index upon vertex removal

2021 ◽  
Vol 6 (12) ◽  
pp. 12976-12985
Author(s):  
Yi Hu ◽  
◽  
Zijiang Zhu ◽  
Pu Wu ◽  
Zehui Shao ◽  
...  
Keyword(s):  
Wiener Index ◽  
Open Problems ◽  
Vertex Removal

<abstract><p>In this paper, we present solutions of two open problems regarding the Wiener index $ W(G) $ of a graph $ G $. More precisely, we prove that for any $ r \geq 2 $, there exist infinitely many graphs $ G $ such that $ W(G) = W(G - \{v_1, \ldots, v_r\}) $, where $ v_1, \ldots, v_r $ are $ r $ distinct vertices of $ G $. We also prove that for any $ r \geq 1 $ there exist infinitely many graphs $ G $ such that $ W(G) = W(G - \{v_i\}) $, $ 1 \leq i \leq r $, where $ v_1, \ldots, v_r $ are $ r $ distinct vertices of $ G $.</p></abstract>

scholarly journals Computing Zagreb, Hyper-Wiener and Degree-Distance indices of four new sums of graphs

2011 ◽  
Vol 27 (2) ◽  
pp. 153-164
Author(s):  
A. R. ASHRAFI ◽  
◽  
A. HAMZEH ◽  
S. HOSSEIN-ZADEH ◽  
◽  
...  
Keyword(s):  
Wiener Index ◽  
Graph Operations ◽  
Degree Distance ◽  
Appl Math

Eliasi and Taeri [M. Eliasi and B. Taeri, Four new sums of graphs and their Wiener indices, Discrete Appl. Math., 157 (2009), 794-803] presented four new sums and computed their Wiener index. In this paper, we continue this work to compute the Zagreb, Hyper-Wiener and Degree-Distance Indices of these graph operations. Some applications are also presented.

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