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The minimum number of spanning trees in regular multigraphs II: the even-degree case

Abstract : In a companion article [The minimum number of spanning trees in regular multigraphs I: the odd-degree case, submitted for publication], we answered a question raised in earlier works by determining the minimum number of spanning trees in a connected d-regular n-vertex multigraph-and identifying all graphs achieving this smallest number-for all odd values of d and all relevant n. The general approach developed there seems relevant to study also all even values of d. However, some additional technicalities need to be dealt with, probably due to the fact the parity of n is not constrained anymore. In particular, the boundary cases (i.e. when d and n are small) show more irregularities than in the odd-degree case, where only two regimes existed regarding the relation between n and d. Specifically, we prove that when n (and d) are even, then every connected d-regular nvertex multigraph G has at least nd/2·(d − 1)^(n/2−1) spanning trees, with equality if and only if G is the even cycle with edges of alternating multiplicities 1 and d − 1, unless d = 4 and n ∈ {6, 8, 10} (in which case the bound is lower, and reached only by the padded paddle graph). If n is odd (and d is even), then every connected d-regular n-vertex multigraph G has at least 1/8·(d − 1)^((n−5)/2)·(3d² − 4d − 4)(d(n − 1) − 2) spanning trees, with equality if and only if G is the fish graph formed by a triangle and an (odd) cycle with edges of multiplicities only 1 and d − 1 that share a single vertex, unless n = 3 or the pair (d, n) belongs to a small number of sporadic values, in which case the (still unique) extremal graph is different.
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https://hal.archives-ouvertes.fr/hal-03472283
Contributor : Jean-Sébastien Sereni Connect in order to contact the contributor
Submitted on : Thursday, December 9, 2021 - 11:44:52 AM
Last modification on : Sunday, June 26, 2022 - 3:23:52 AM

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Jakub Pekárek, Jean-Sébastien Sereni, Zelealem B yilma. The minimum number of spanning trees in regular multigraphs II: the even-degree case. 2021. ⟨hal-03472283⟩

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