On the Number of Nonisomorphic Subtrees of a Tree

Czabarka, Éva; Székely, László Á.; Wagner, Stephan

doi:10.1002/jgt.22144

Cited by 14 publications

(4 citation statements)

References 7 publications

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“…Therefore, algorithms that only target specific subgraphs are not considered (e.g., triads [163], cliques [7,51], stars [37,54] or subtrees [91]). Furthermore, given our focus on generalizability, we do not consider algorithms that are only capable of counting sugraphs in specific graphs (e.g., bipartite networks [161], trees [36]), or that only count local subgraphs [38].…”

Section: Algorithms Not Consideredmentioning

confidence: 99%

A Survey on Subgraph Counting: Concepts, Algorithms and Applications to Network Motifs and Graphlets

Ribeiro,

Paredes,

Silva

et al. 2019

Preprint

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Computing subgraph frequencies is a fundamental task that lies at the core of several network analysis methodologies, such as network motifs and graphlet-based metrics, which have been widely used to categorize and compare networks from multiple domains. Counting subgraphs is however computationally very expensive and there has been a large body of work on efficient algorithms and strategies to make subgraph counting feasible for larger subgraphs and networks.This survey aims precisely to provide a comprehensive overview of the existing methods for subgraph counting. Our main contribution is a general and structured review of existing algorithms, classifying them on a set of key characteristics, highlighting their main similarities and differences. We identify and describe the main conceptual approaches, giving insight on their advantages and limitations, and provide pointers to existing implementations. We initially focus on exact sequential algorithms, but we also do a thorough survey on approximate methodologies (with a trade-off between accuracy and execution time) and parallel strategies (that need to deal with an unbalanced search space).

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Section: Algorithms Not Consideredmentioning

confidence: 99%

A Survey on Subgraph Counting: Concepts, Algorithms and Applications to Network Motifs and Graphlets

Ribeiro,

Paredes,

Silva

et al. 2019

Preprint

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show abstract

“…The subtree number index STN(G) of a graph G is a structure-based index, defined as the total number of non-empty subtrees of G. It is discovered to have applications in the design of reliable communication network [21], bioinformatics [11], and characterizing physicochemical and structural properties of molecular graphs [13,26,25]. In recent years there have been related works on enumerating subtrees [22,15,3,2,28], characterizing extremal graphs and values [16,29,10,30], analyzing relations with other topological indices such as the Wiener index [26,25,17,19], average order and density of subtrees [18,9,6].…”

Section: Introductionmentioning

confidence: 99%

The expected subtree number index in random polyphenylene and spiro chains

Sun

Jun

et al. 2020

Discrete Applied Mathematics

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The subtree number index STN(G) of a simple graph G is the number of nonempty subtrees of G. It is a structural and counting topological index that has received more and more attention in recent years. In this paper we first obtain exact formulas for the expected values of subtree number index of random polyphenylene and spiro chains, which are molecular graphs of a class of unbranched multispiro molecules and polycyclic aromatic hydrocarbons. Moreover, we establish a relation between the expected values of the subtree number indices of a random polyphenylene and its corresponding hexagonal squeeze. We also present the average values for subtree number indices with respect to the set of all polyphenylene and spiro chains with n hexagons.

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“…The aim of this paper is to present the corresponding Nordhaus-Gaddum type inequalities for the number of connected induced subgraphs in a graph. Let η(G) be the number of nonempty connected induced subgraphs in a graph G. If G is acyclic, then η(G) counts precisely the number of subtrees in G. The number of subtrees is a fairly popular graph invariant since it has been studied in several contexts; see for example [2,13,18,19,25,26,27,28] and the recent papers [5,6,8,29]. On the other hand, the number of connected subgraphs or connected induced subgraphs (especially extremal problems) is only starting to attract attention of researchers [1,20,22].…”

Section: Introductionmentioning

confidence: 99%

Nordhaus-Gaddum inequalities for the number of connected induced subgraphs in graphs

Andriantiana¹,

Dossou-Olory²

2020

Preprint

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Let η(G) be the number of connected induced subgraphs in a graph G, and G the complement of G. We prove that η(G) + η(G) is minimum, among all n-vertex graphs, if and only if G has no induced path on four vertices. Since the n-vertex tree S n with maximum degree n − 1 is the unique tree of diameter 2, η(S n ) + η(S n ) is minimum among all n-vertex trees, while the maximum is shown to be achieved only by the tree whose degree sequence is ( n/2 , n/2 , 1, . . . , 1). Furthermore, we prove that every graph G of order n ≥ 5 and with maximum η(G) + η(G) must have diameter at most 3, no cut vertex and the property that G is also connected. In both cases of trees and graphs of fixed order, we find that if η(G) is maximum then η(G) + η(G) is minimum.As a corollary to our results, we characterise the unique graph G of given order and number of pendent vertices, and the unique unicyclic graph G of a given order that minimises η(G) + η(G).

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On the Number of Nonisomorphic Subtrees of a Tree

Cited by 14 publications

References 7 publications

A Survey on Subgraph Counting: Concepts, Algorithms and Applications to Network Motifs and Graphlets

A Survey on Subgraph Counting: Concepts, Algorithms and Applications to Network Motifs and Graphlets

The expected subtree number index in random polyphenylene and spiro chains

Nordhaus-Gaddum inequalities for the number of connected induced subgraphs in graphs

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