Modularity of minor‐free graphs

Sułkowska, Małgorzata; Lasoń, Michał

doi:10.1002/jgt.22896

Cited by 2 publications

(2 citation statements)

References 25 publications

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“…These results are true also in the weighted setting where each vertex x ∈ V (G) is assigned a weight w(x) with 0 w(x) 1 2 , the total weight of all vertices is 1, and the edge separator should split G into components of weight at most 1 2 . Lasoń and Sulkowska [6] showed that in the weighted setting, if G is an n-vertex K t -minor-free graph of maximum degree ∆ = o(n) and the vertices are weighted proportionally to their degrees, then there exists a balanced edge separator of size o(n). Their proof relies on spectral methods-more precisely, on an upper bound for the second smallest eigenvalue of the Laplacian matrix of K t -minor-free graphs due to Biswal, Lee, and Rao [2]-and only works for these specific weights.…”

Section: Introductionmentioning

confidence: 99%

Edge Separators for Graphs Excluding a Minor

Joret,

Lochet,

Seweryn

2023

Electron. J. Combin.

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We prove that every $n$-vertex $K_t$-minor-free graph $G$ of maximum degree $\Delta$ has a set $F$ of $O(t^2(\log t)^{1/4}\sqrt{\Delta n})$ edges such that every component of $G - F$ has at most $n/2$ vertices. This is best possible up to the dependency on $t$ and extends earlier results of Diks, Djidjev, Sýkora, and Vrťo (1993) for planar graphs, and of Sýkora and Vrťo (1993) for bounded-genus graphs. Our result is a consequence of the following more general result: The line graph of $G$ is isomorphic to a subgraph of the strong product $H \boxtimes K_{\lfloor p \rfloor}$ for some graph $H$ with treewidth at most $t-2$ and $p = \sqrt{(t-3)\Delta |E(G)|} + \Delta$.

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Section: Introductionmentioning

confidence: 99%

Edge Separators for Graphs Excluding a Minor

Joret,

Lochet,

Seweryn

2023

Electron. J. Combin.

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“…For very recent results on the modularity for random graphs (3-regular ones, the ones with a given degree sequence and the ones on the hyperbolic plane) consult Chellig et al (2021), Chellig et al (2022). Also, for a new result on the modularity of minor-free graphs one may check Lasoń & Sulkowska (2022).…”

Section: Introductionmentioning

confidence: 99%

Preferential attachment hypergraph with high modularity

Giroire

Nisse

Trolliet

et al. 2022

Net Sci

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Numerous works have been proposed to generate random graphs preserving the same properties as real-life large-scale networks. However, many real networks are better represented by hypergraphs. Few models for generating random hypergraphs exist, and also, just a few models allow to both preserve a power-law degree distribution and a high modularity indicating the presence of communities. We present a dynamic preferential attachment hypergraph model which features partition into communities. We prove that its degree distribution follows a power-law, and we give theoretical lower bounds for its modularity. We compare its characteristics with a real-life co-authorship network and show that our model achieves good performances. We believe that our hypergraph model will be an interesting tool that may be used in many research domains in order to reflect better real-life phenomena.

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Modularity of minor‐free graphs

Abstract: We prove that a class of graphs with excluded minor and with the maximum degree of smaller order than the number of edges is maximally modular, that is, for every ε > 0, the modularity of any graph in the class with sufficiently many edges is at least ε 1 − .

Cited by 2 publications

References 25 publications

Edge Separators for Graphs Excluding a Minor

Edge Separators for Graphs Excluding a Minor

Preferential attachment hypergraph with high modularity

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