Upper bounds for the number of spanning trees of graphs

Bozkurt, Ş. Burcu

doi:10.1186/1029-242x-2012-269

Cited by 7 publications

(8 citation statements)

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“…It may be remarked that Corollary 5.3 can be proved using Proposition 5.5 and induction as well. The following bound has been obtained in [6].…”

Section: Maximizing the Number Of Spanning Trees In A Bipartite Graphmentioning

confidence: 99%

Maximizing Spectral Radius and Number of Spanning Trees in Bipartite Graphs

Bapat

2018

Indian Statistical Institute Series

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The problems of maximizing the spectral radius and the number of spanning trees in a class of bipartite graphs with certain degree constraints are considered. In both the problems, the optimal graph is conjectured to be a Ferrers graph. Known results towards the resolution of the conjectures are described. We give yet another proof of a formula due to Ehrenborg and van Willigenburg for the number of spanning trees in a Ferrers graph. The main tool is a result which gives several necessary and sufficient conditions under which the removal of an edge in a graph does not affect the resistance distance between the end-vertices of another edge.

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“…It may be remarked that Corollary 5.3 can be proved using Proposition 5.5 and induction as well. The following bound has been obtained in [6].…”

Section: Maximizing the Number Of Spanning Trees In A Bipartite Graphmentioning

confidence: 99%

Maximizing Spectral Radius and Number of Spanning Trees in Bipartite Graphs

Bapat

2018

Indian Statistical Institute Series

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“…The key idea behind the bound above is to apply the AM-GM inequality to the normalized eigenvalues. Other recent advances in constructing upper bounds for the number of spanning trees can be found in [9], another overview is given in [8].…”

Section: Introductionmentioning

confidence: 99%

“…Though, for a bipartite graph we know some improvement of the bound (1) . For example, in [8] the following upper bound was shown for a bipartite connected graph…”

Section: Introductionmentioning

confidence: 99%

On the number of spanning trees in bipartite graphs

Volkova

2020

Preprint

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In this paper, we address the Ehrenborg's conjecture which proposes that for any bipartite graph G the inequality τ (G) ≤ m −1 n −1 D(G) holds. Here τ (G) denotes the number of spanning trees of G, D(G) is the product of the degrees of the vertices and m, n are the sizes of the components. We show that the conjecture is true for a one-side regular graph (that is a graph for which all degrees of the vertices of at least one of the components are equal). We also present a proof alternative to one given in [7] of the fact that the equality holds for Ferrers graphs.

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“…Now, we give some known upper bounds on t ( G ): Grimmett [ 11 ]:

Grone and Merris [ 12 ]:

Nosal [ 13 ]: for r -regular graphs,

Cvetković et al (see [ 5 , page 222]):

where

is the number of edges of

, Das [ 14 ]:

Zhang [ 15 ]:

where a = (( n ( n −1) − 2 m )/2 mn ( n −2)) 1/2 , Feng et al [ 16 ]:

Li et al [ 17 ]:

Bozkurt [ 18 ]:

where b = (( n −1−Δ 1 )/ n ( n −2)Δ 1 ) 1/2 , Das et al [ 19 ]:

…”

Section: Introductionmentioning

confidence: 99%

“…However, Das et al [ 19 ] proved that ( 11 ) is not true for K n . In [ 15 , 16 , 18 ] the authors showed that ( 8 ) is better than ( 3 ), ( 9 ) is better than ( 7 ) and ( 10 ), and ( 12 ) is better than ( 4 ). For more bounds and the relations between the number of spanning trees and the structural parameters of graphs such as connectivity, chromatic number, independence number, and clique number, see [ 17 , 19 ].…”

Section: Introductionmentioning

confidence: 99%