A Common Asymptotic Behavior for Different Classes of Sparse Labelled Graphs with Given Number of Vertices and Edges

Vlady Ravelomanana () and Loÿs Thimonier ()
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Vlady Ravelomanana: LaRIA
Loÿs Thimonier: LaRIA

A chapter in Formal Power Series and Algebraic Combinatorics, 2000, pp 309-319 from Springer

Abstract: Abstract Let m(n, n + k) be the number of connected labelled multi-graphs, which are graphs with n vertices, n + k edges and possible self-loops and/or multiple edges. Denote by c(n, n + k) the number of connected labelled simple graphs with the same parameters. First, under the condition that k = o(n 2), by making use of the methods developped by Bender et al. in [3], we show that m(n, n + k) ≈ c(n,n + k) as n → ∞. Under the same condition on the number of exceeding edges, k = o(n 2), these results are extended to show that connected labelled graphs, multi-graphs and graphs without a finite set of forbidden subgraphs have the same asymptotic behavior. Finally, we give sufficient condition, in terms of the total number of graphs, for the probability of connectedness to have a limit equal to 1 as the number of vertices tends to ∞.

Date: 2000
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DOI: 10.1007/978-3-662-04166-6_28

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