Problems related to graph indices in trees

Székely, László Á.; Wagner, Stephen; Wang, Hua

doi:10.1007/978-3-319-24298-9_1

Cited by 5 publications

(3 citation statements)

References 115 publications

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“…Common generalizations of this optimization problem for the undirected and unweighted case include the dense, sparse, and minimum routing cost spanning tree problems. Many other variations of optimal spanning trees exist [15,16,[27][28][29][30][31][32][33][34][35][36], and such trees are relevant to an ever widening diversity of applications from optical network design [14,37] to networked oscillator synchronization [38].…”

Section: Introductionmentioning

confidence: 99%

Spanning Trees of Recursive Scale-Free Graphs

Diggans,

Bollt,

ben-Avraham

2021

Preprint

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We present a link-by-link rule-based method for constructing all members of the ensemble of spanning trees for any recursively generated, finitely articulated graph, such as the DGM net. The recursions allow for many large-scale properties of the ensemble of spanning trees to be analytically solved exactly. We show how a judicious application of the prescribed growth rules selects for certain subsets of the spanning trees with particular desired properties (small-world or extended diameter, degree distribution, etc.), and thus provides solutions to several optimization problems. The analysis of spanning trees enhances the usefulness of recursive graphs as sophisticated models for everyday life complex networks.

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Section: Introductionmentioning

confidence: 99%

Spanning Trees of Recursive Scale-Free Graphs

Diggans,

Bollt,

ben-Avraham

2021

Preprint

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“…where t w is the subtree of t above the vertex w and rooted at w. Such functionals are encountered in computer science where they represent the cost of divide-and-conquer algorithms, in phylogenetics where they are used as a rough measure of tree shape to detect imbalance or in chemical graph theory where they appear as a predictive tool for some chemical properties. Among these, we mention the total path length defined as the sum of the distances to the root of all vertices, the Wiener index [43] defined as the sum of the distances between all pairs of vertices, the shape functional, the Sackin index, the Colless index and the cophenetic index, see [42] for their definitions and also [14] for their representation using additive functionals, and the references therein. See also [39] for other functionals such that the number of matchings, dominating sets, independent sets for trees.…”

Section: Introductionmentioning

confidence: 99%

Global Regime for General Additive Functionals of Conditioned Bienaym{é}-Galton-Watson Trees

Abraham¹,

Delmas²,

Nassif³

2020

Preprint

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We give an invariance principle for very general additive functionals of conditioned Bienaymé-Galton-Watson trees in the global regime when the offspring distribution lies in the domain of attraction of a stable distribution, the limit being an additive functional of a stable Lévy tree. This includes the case when the offspring distribution has finite variance (the Lévy tree being then the Brownian tree). We also describe, using an integral test, a phase transition for toll functions depending on the size and height.

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“…Extremal trees and graphs that minimize the Wiener index in various classes of graphs have been extensively studied, see [ 8 ] for an earlier informative survey and part of [ 9 ] for some recent results. One interesting observation was that the extremal structures that minimize the Wiener index usually maximize the number of subtrees (see for instance [ 10 ]).…”

Section: Introductionmentioning

confidence: 99%

Degree sums and dense spanning trees

Gao

Dong

et al. 2017

PLoS ONE

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Finding dense spanning trees (DST) in unweighted graphs is a variation of the well studied minimum spanning tree problem (MST). We utilize established mathematical properties of extremal structures with the minimum sum of distances between vertices to formulate some general conditions on the sum of vertex degrees. We analyze the performance of various combinations of these degree sum conditions in finding dense spanning subtrees and apply our approach to practical examples. After briefly describing our algorithm we also show how it can be used on variations of DST, motivated by variations of MST. Our work provide some insights on the role of various degree sums in forming dense spanning trees and hopefully lay the foundation for finding fast algorithms or heuristics for related problems.

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Problems related to graph indices in trees

Cited by 5 publications

References 115 publications

Spanning Trees of Recursive Scale-Free Graphs

Spanning Trees of Recursive Scale-Free Graphs

Global Regime for General Additive Functionals of Conditioned Bienaym{é}-Galton-Watson Trees

Degree sums and dense spanning trees

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