2024/06/17 by Qingying Deng, Deng, Qingying, Xian’an Jin +3 · 2 citations
Mathematics · Engineering · #Mathematics and Applications #Mathematical Dynamics and Fractals #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2406.11648
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.