The number of quasi-trees of bouquets with exactly one non-orientable loop
Abstract
Recently, Merino extended the classical relation between the $2n$-th Fibonacci number and the number of spanning trees of the $n$-fan graph to ribbon graphs, and established a relation between the $n$-associated Mersenne number and the number of quasi-trees of the $n$-wheel ribbon graph. Moreover, Merino posed a problem of finding the Lucas numbers as the number of spanning quasi-trees of a family of ribbon graphs. In this paper, we solve the problem and give the Matrix-Quasi-tree Theorem for a bouquet with exactly one non-orientable loop. Furthermore, this theorem is used to verify that the number of quasi-trees of some classes of bouquets is closely related to the Fibonacci and Lucas numbers. We also give alternative proofs of the number of quasi-trees of these bouquets by using the deletion-contraction relations of ribbon graphs.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.11648
- arXiv:
- arXiv:2406.11648
- Bibcode:
- 2024arXiv240611648D
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- 22 pages, 3 figures