Random Graphs from a Weighted Minor-Closed Class

McDiarmid, Colin

doi:10.37236/2793

Cited by 9 publications

(17 citation statements)

References 62 publications

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“…Before we introduce the asymptotic results for a general bridge-alterable set of graphs, let us record some results on random forests R n ∈ τ F which generalise the results mentioned earlier for uniform random forests R n ∈ u F -see for example [10] where these results are proved in a general setting. Observe that τ (F ) = f (K n )(λ/ν) e(F ) ν n for each F ∈ F n (whereK n denotes the graph on [n] with no edges): thus τ (F ) ∝ (λ/ν) e(F ) , and the only aspect of the weighting that matters is the ratio λ/ν.…”

Section: Random Weighted Graphsmentioning

confidence: 99%

Connectivity for Random Graphs from a Weighted Bridge-Addable Class

McDiarmid

2012

Electron. J. Combin.

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There has been much recent interest in random graphs sampled uniformly from the n-vertex graphs in a suitable structured class, such as the class of all planar graphs. Here we consider a general bridgeaddable class A of graphs -if a graph is in A and u and v are vertices in different components then the graph obtained by adding an edge (bridge) between u and v must also be in A. Various bounds are known concerning the probability of a random graph from such a class being connected or having many components, sometimes under the additional assumption that bridges can be deleted as well as added.Here we improve or amplify or generalise these bounds (though we do not resolve the main conjecture). For example, we see that the expected number of vertices left when we remove a largest component is less than 2. The generalisation is to consider 'weighted' random graphs, sampled from a suitable more general distribution, where the focus is on the bridges.

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Section: Random Weighted Graphsmentioning

confidence: 99%

Connectivity for Random Graphs from a Weighted Bridge-Addable Class

McDiarmid

2012

Electron. J. Combin.

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“…generalizing and improving Lemma 2.6 in [10]. For F n ∈ u F, where F is the class of forests, we know that E[frag(F n )] → 1 (see for example [12]), which leads us to the next conjecture, extending Conjecture 1.1.…”

Section: Introductionmentioning

confidence: 62%

“…Proof. We apply results on weighted random graphs from [12]. Let τ = (μ, ν), where μ > 0 is a constant edge-weighting and ν > 0 is a component-weighting.…”

Section: Proof Of Theorem 31 Part (C)mentioning

confidence: 99%

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Bridge-Addability, Edge-Expansion and Connectivity

McDiarmid¹,

Weller

2017

Combinator. Probab. Comp.

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A class of graphs is called bridge-addable if, for each graph in the class and each pair u and v of vertices in different components, the graph obtained by adding an edge joining u and v must also be in the class. The concept was introduced in 2005 by McDiarmid, Steger and Welsh, who showed that, for a random graph sampled uniformly from such a class, the probability that it is connected is at least 1/e.We generalize this and related results to bridge-addable classes with edge-weights which have an edge-expansion property. Here, a graph is sampled with probability proportional to the product of its edge-weights. We obtain for example lower bounds for the probability of connectedness of a graph sampled uniformly from a relatively bridge-addable class of graphs, where some but not necessarily all of the possible bridges are allowed to be introduced. Furthermore, we investigate whether these bounds are tight, and in particular give detailed results about random forests in complete balanced multipartite graphs.

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“…We weigh the graphs by attaching weights to each edge. There is an extensive literature on properties of weighted graphs (where we may weight either the edges or the graphs in the family); see [ALHM,AL,Bo1,Bo2,ES,Ga,McD1,McD2,Po] and the references therein for some results and applications.…”

Section: Introductionmentioning

confidence: 99%

On the spectral distribution of large weighted random regular graphs

Goldmakher

Khoury

Miller

et al. 2014

Random Matrices: Theory Appl.

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ABSTRACT. McKay proved the limiting spectral measures of the ensembles of d-regular graphs with N vertices converge to Kesten's measure as N → ∞. Given a large d-regular graph we assign random weights, drawn from some distribution W, to its edges. We study the relationship between W and the associated limiting spectral distribution obtained by averaging over the weighted graphs. We establish the existence of a unique 'eigendistribution' (a weight distribution W such that the associated limiting spectral distribution is a rescaling of W). Initial investigations suggested that the eigendistribution was the semi-circle distribution, which by Wigner's Law is the limiting spectral measure for real symmetric matrices. We prove this is not the case, though the deviation between the eigendistribution and the semi-circular density is small (the first seven moments agree, and the difference in each higher moment is O(1/d2 )). Our analysis uses combinatorial results about closed acyclic walks in large trees, which may be of independent interest.

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Random Graphs from a Weighted Minor-Closed Class

Cited by 9 publications

References 62 publications

Connectivity for Random Graphs from a Weighted Bridge-Addable Class

Connectivity for Random Graphs from a Weighted Bridge-Addable Class

Bridge-Addability, Edge-Expansion and Connectivity

On the spectral distribution of large weighted random regular graphs

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