Subtrees of graphs

Chin, A. H.; Gordon, Gary; MacPhee, Kellie J.; Vincent, C.H.

doi:10.1002/jgt.22359

Cited by 30 publications

(40 citation statements)

References 15 publications

(21 reference statements)

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“…Similarly, as studied in paper [35], we can also discuss the ratio of spanning trees to all subtrees and spanning tree densities of K j 1,n and W n . We skip the details here.…”

Section: Subtree Densities Of K J 1n and W Nmentioning

confidence: 94%

“…1,j−r ; f , 1; c 0 ) and F(K 0 1,0 ; f , 1; c 0 ) = y, F(K 1 1,1 ; f , 1; c 0 ) = y + y 2 . From Equations (33)- (35), we can obtain the subtree densities of K j 1,n (1 ≤ j ≤ n) (plotted in Figure 6); related data can be found in Table 2. Similarly, from Equations (33), (34), and (36), we have subtree densities W n plotted in Figure 7; related data are listed in Table 3.…”

Section: Subtree Densities Of K J 1n and W Nmentioning

confidence: 99%

“…With structure mapping and weights transferring of cycles, Yang et al [34] also presented enumerating algorithms for subtrees of hexagonal and phenylene chains. More recently, Chin et al [35] presented the subtree number of complete graphs, complete bipartite graphs, and theta graphs, as well as the ratio of spanning trees to all subtrees of the above graphs.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Subtrees of Fan Graphs, Wheel Graphs, and “Partitions” of Wheel Graphs under Dynamic Evolution

Wang

et al. 2019

Mathematics

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The number of subtrees, or simply the subtree number, is one of the most studied counting-based graph invariants that has applications in many interdisciplinary fields such as phylogenetic reconstruction. Motivated from the study of graph surgeries on evolutionary dynamics, we consider the subtree problems of fan graphs, wheel graphs, and the class of graphs obtained from “partitioning” wheel graphs under dynamic evolution. The enumeration of these subtree numbers is done through the so-called subtree generation functions of graphs. With the enumerative result, we briefly explore the extremal problems in the corresponding class of graphs. Some interesting observations on the behavior of the subtree number are also presented.

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“…Similarly, as studied in paper [35], we can also discuss the ratio of spanning trees to all subtrees and spanning tree densities of K j 1,n and W n . We skip the details here.…”

Section: Subtree Densities Of K J 1n and W Nmentioning

confidence: 94%

Section: Subtree Densities Of K J 1n and W Nmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Subtrees of Fan Graphs, Wheel Graphs, and “Partitions” of Wheel Graphs under Dynamic Evolution

Wang

et al. 2019

Mathematics

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“…Recently, Chin et al [2] initiated the study of subtrees of multigraphs in general. For a given multigraph

G

, two parameters introduced by Chin et al are the mean subtree order of

G

, denoted

μ (G)

, and the proportion of subtrees of

G

that are spanning, denoted

P (G)

.…”

Section: Introductionmentioning

confidence: 99%

“…In this note, we are concerned with the following conjecture of Chin et al [2], and some related problems.…”

Section: Introductionmentioning

confidence: 99%

On the mean subtree order of graphs under edge addition

Cameron

Mol

2020

Journal of Graph Theory

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For a graph G, the mean subtree order of G is the average order of a subtree of G. In this note, we provide counterexamples to a recent conjecture of Chin, Gordon, MacPhee, and Vincent, that for every connected graph G and every pair of distinct vertices u and v of G, the addition of the edge between u and v increases the mean subtree order. In fact, we show that the addition of a single edge between a pair of nonadjacent vertices in a graph of order n can decrease the mean subtree order by as much as n∕3 asymptotically. We propose the weaker conjecture that for every connected graph G which is not complete, there exists a pair of nonadjacent vertices u and v, such that the addition of the edge between u and v increases the mean subtree order. We prove this conjecture in the special case that G is a tree.

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On the probability that a random subtree is spanning

Wagner

2021

Journal of Graph Theory

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We consider the quantity P ( G ) associated with a graph G that is defined as the probability that a randomly chosen subtree of G is spanning. Motivated by conjectures due to Chin, Gordon, MacPhee and Vincent on the behaviour of this graph invariant depending on the edge density, we establish first that P ( G ) is bounded below by a positive constant provided that the minimum degree is bounded below by a linear function in the number of vertices. Thereafter, the focus is shifted to the classical Erdős–Rényi random graph model G ( n , p ). It is shown that P ( G ) converges in probability to e − 1 ∕ ( e p ∞ ) if p → p ∞ > 0 and to 0 if p → 0. As side results, we find in the dense case that the total number of subtrees satisfies a log‐normal limit law, and that the number of vertices missing in a random subtree asymptotically follows a Poisson distribution.

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Subtrees of graphs

Cited by 30 publications

References 15 publications

On Subtrees of Fan Graphs, Wheel Graphs, and “Partitions” of Wheel Graphs under Dynamic Evolution

On Subtrees of Fan Graphs, Wheel Graphs, and “Partitions” of Wheel Graphs under Dynamic Evolution

On the mean subtree order of graphs under edge addition

On the probability that a random subtree is spanning

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