On the Minimum Number of Spanning Trees in Cubic Multigraphs
Let G2n, H2n be two non-isomorphic connected cubic multigraphs of order 2n with parallel edges permitted but without loops. Let t(G2n), t (H2n) denote the number of spanning trees in G2n, H2n, respectively. We prove that for n ≥ 3 there is the unique G2n such that t(G2n) < t(H2n) for any H2n. Fur...
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| Format: | Article |
| Language: | English |
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Sciendo
2020-02-01
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| Series: | Discussiones Mathematicae Graph Theory |
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| Online Access: | https://doi.org/10.7151/dmgt.2123 |
| Summary: | Let G2n, H2n be two non-isomorphic connected cubic multigraphs of order 2n with parallel edges permitted but without loops. Let t(G2n), t (H2n) denote the number of spanning trees in G2n, H2n, respectively. We prove that for n ≥ 3 there is the unique G2n such that t(G2n) < t(H2n) for any H2n. Furthermore, we prove that such a graph has t(G2n = 522n−3 spanning trees. Based on our results we give a conjecture for the unique r-regular connected graph H2n of order 2n and odd degree r that minimizes the number of spanning trees. |
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| ISSN: | 2083-5892 |