Asymmetric circular graph with Hosoya index and negative continued fractions

It has been known that the Hosoya index of caterpillar graph can be calculated as the numerator of the simple continued fraction. Recently in [MATCH Commun. Math. Comput. Chem. 2020, 84 (2), 399-428], the author introduces a more general graph called caterpillar-bond graph and shows that its Hosoya index can be calculated as the numerator of the general continued fraction. In this paper, we show how the Hosoya index of the graph with non-uniform ring structure can be calculated from the negative continued fraction. We also give the relation between some radial graphs and multidimensional continued fractions in the sense of the Hosoya index.

Simultaneous approximations to p-adic numbers and algebraic dependence via multidimensional continued fractions

Unlike the real case, there are not many studies and general techniques for providing simultaneous approximations in the field of p-adic numbers $$\mathbb Q_p$$ Q p . Here, we study the use of multidimensional continued fractions (MCFs) in this context. MCFs were introduced in $$\mathbb R$$ R by Jacobi and Perron as a generalization of continued fractions and they have been recently defined also in $$\mathbb Q_p$$ Q p . We focus on the dimension two and study the quality of the simultaneous approximation to two p-adic numbers provided by p-adic MCFs, where p is an odd prime. Moreover, given algebraically dependent p-adic numbers, we see when infinitely many simultaneous approximations satisfy the same algebraic relation. This also allows to give a condition that ensures the finiteness of the p-adic Jacobi–Perron algorithm when it processes some kinds of $$\mathbb Q$$ Q -linearly dependent inputs.

Generalizing the Minkowski question mark function to a family of multidimensional continued fractions

The Minkowski question mark function [Formula: see text] is a continuous, strictly increasing, one-to-one and onto function that has derivative zero almost everywhere. Key to these facts are the basic properties of continued fractions. Thus [Formula: see text] is a naturally occurring number theoretic singular function. This paper generalizes the question mark function to the 216 triangle partition (TRIP) maps. These are multidimensional continued fractions which generate a family of almost all known multidimensional continued fractions. We show for each TRIP map that there is a natural candidate for its analog of the Minkowski question mark function. We then show that the analog is singular for 96 of the TRIP maps and show that 60 more are singular under an assumption of ergodicity.