Abstract
Although the results above seem to yield good estimators of an upper bound on connectivity, they can easily give rather poor estimates of k itself. One possible improvement could be effected by using the theorem of Harary and Chartrand [2] that δ≥p−2+n / 2 for some n such that 1≤n≤p−1 implies k≧n. This could give an estimate of a lower bound on k by using δ* in place of δ in the above inequality. Of course, the usefulness of this is limited to cases in which δ* is rather large, at least 1/2 p.
Further approaches to this problem based on testing the hypothesis that k=1 will hopefully be the subject of a future paper.
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References
M. Capobianco, Statistical inference in finite populations having structure, Trans. New York Acad. Sci. 32 (1970), 401–143.
G. Chartrand and F. Harary, Graphs with prescribed connectivities, Theory of Graphs, (P. Erdös and G. Katona, Eds.) Akademiai Kiado, Budapest, (1968), 61–63.
P. Zweig Chinn, The frequency partition of a graph, Recent Trends in Graph Theory, (M. Capobianco, J. Frechen, M. Krolik, Eds.), Springer-Verlag, (1971).
F. Harary, The maximum connectivity of a graph, Proc. Nat. Acad. Sci. USA 48 (1962), 1142–1146.
H. Whitney, Congruent graphs and the connectivity of graphs, Amer. J. Math. 54 (1932), 150–168.
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Capobianco, M. (1972). Estimating the connectivity of a graph. In: Alavi, Y., Lick, D.R., White, A.T. (eds) Graph Theory and Applications. Lecture Notes in Mathematics, vol 303. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0067358
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DOI: https://doi.org/10.1007/BFb0067358
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