The number of quasi-trees of bouquets with exactly one non-orientable loop

TL;DR

This paper establishes a new theorem relating the number of quasi-trees in bouquets with one non-orientable loop to Fibonacci and Lucas numbers, solving a problem posed by Merino and providing alternative proofs.

Contribution

It introduces the Matrix-Quasi-tree Theorem for bouquets with a single non-orientable loop and links quasi-tree counts to Fibonacci and Lucas numbers.

Findings

Number of quasi-trees relates to Fibonacci and Lucas numbers.

Provides the Matrix-Quasi-tree Theorem for specific ribbon graphs.

Offers alternative proofs using deletion-contraction relations.

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.