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Long cycles and neighborhood union in 1-tough graphs with large degree sums

100%

Hoa V.

Discussiones Mathematicae Graph Theory

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1998

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tom 18

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nr 1

5-13

EN

For a 1-tough graph G we define σ₃(G) = min{d(u) + d(v) + d(w):{u,v,w} is an independent set of vertices} and $NC_{σ₃-n+5}(G)$ = $max{⋃_{i = 1}^{σ₃-n+5}$ $N(v_i) : {v₁, ..., v_{σ₃-n+5}}$ is an independent set of vertices}. We show that every 1-tough graph with σ₃(G) ≥ n contains a cycle of length at least $min{n,2NC_{σ₃-n+5}(G)+2}$. This result implies some well-known results of Faßbender [2] and of Flandrin, Jung & Li [6]. The main result of this paper also implies that c(G) ≥ min{n,2NC₂(G)+2} where NC₂(G) = min{|N(u) ∪ N(v)|:d(u,v) = 2}. This strengthens a result that c(G) ≥ min{n, 2NC₂(G)} of Bauer, Fan and Veldman [3].

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Local dependency in networks

100%

Kudělka M., Zehnalová Š., Horák Z., Krömer P., Snášel V.

International Journal of Applied Mathematics and Computer Science

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2015

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tom 25

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nr 2

281-293

EN

Many real world data and processes have a network structure and can usefully be represented as graphs. Network analysis focuses on the relations among the nodes exploring the properties of each network. We introduce a method for measuring the strength of the relationship between two nodes of a network and for their ranking. This method is applicable to all kinds of networks, including directed and weighted networks. The approach extracts dependency relations among the network's nodes from the structure in local surroundings of individual nodes. For the tasks we deal with in this article, the key technical parameter is locality. Since only the surroundings of the examined nodes are used in computations, there is no need to analyze the entire network. This allows the application of our approach in the area of large-scale networks. We present several experiments using small networks as well as large-scale artificial and real world networks. The results of the experiments show high effectiveness due to the locality of our approach and also high quality node ranking comparable to PageRank.

3

Tchebotaröv’s extremal problem

63%

Tamrazov P.

Open Mathematics

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2005

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tom 3

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nr 4

591-605

EN

We give the complete solution of the extremal problem posed by N.G. Tchebotaröv in 20th of the last century, and we establish explicit parametric formulae for the extremals.

Ograniczanie wyników

1 Discussiones Mathematicae Graph Theory

1 International Journal of Applied Mathematics and Computer Science

1 Open Mathematics

1 Hoa V.

1 Horák Z.

1 Krömer P.

1 Kudělka M.

1 Snášel V.

1 Tamrazov P.

1 Zehnalová Š.

1 2015

1 2005

1 1998

Long cycles and neighborhood union in 1-tough graphs with large degree sums

Local dependency in networks

Tchebotaröv’s extremal problem