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The minimum number of spanning trees in regular multigraphs I: the odd-degree case

Abstract : In a recent article, Bogdanowicz determines the minimum number of spanning trees a connected cubic multigraph on a fixed number of vertices can have and identifies the unique graph that attains this minimum value. He conjectures that a generalized form of this construction, which we here call a padded paddle graph, would be extremal for d-regular multigraphs where d ≥ 5 is odd. We prove that, indeed, the padded paddle minimises the number of spanning trees, but this is true only when the number of vertices, n, is greater than (9d+6)/8. We show that a different graph, which we here call the padded cycle, is optimal for n < (9d+6)/8. This fully determines the d-regular multi-graphs minimising the number of spanning trees for odd values of d. The approach we develop can also be applied to the even-degree case. However, the extremal structures are more irregular, and the slightly more technical analysis is done in a companion article.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03472264
Contributor : Jean-Sébastien Sereni Connect in order to contact the contributor
Submitted on : Thursday, December 9, 2021 - 11:37:24 AM
Last modification on : Sunday, June 26, 2022 - 3:23:52 AM

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  • HAL Id : hal-03472264, version 1

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Jakub Pekárek, Jean-Sébastien Sereni, Zelealem B yilma. The minimum number of spanning trees in regular multigraphs I: the odd-degree case. 2021. ⟨hal-03472264⟩

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