Related papers: Golden ratio in graph theory: A survey
Sarnak's golden mean conjecture states that $(m+1)d_\varphi(m)\le1+\frac{2}{\sqrt{5}}$ for all integers $m\ge1$, where $\varphi$ is the golden mean and $d_\theta$ is the discrepancy function for $m+1$ multiples of $\theta$ modulo 1. In this…
A graph $G$ is defined encapsulating the number theoretic notion of the Fundamental Theorem of Arithmetic. We then provide a graph theoretic approach to the fundamental results on the coprimality of two natural numbers, through the use of…
We study the relation between the growth rate of a graph property and the entropy of the graph limits that arise from graphs with that property. In particular, for hereditary classes we obtain a new description of the colouring number,…
In this paper, we introduce a connection between two classical concepts of graph theory: \; metric dimension and distinguishing number. For a given graph $G$, let ${\rm dim}(G)$ and $D(G)$ represent its metric dimension and distinguishing…
An amusing connection between Ford circles, Fibonacci numbers, and golden ratio is shown. Namely, certain tangency points of Ford circles are concyclic and involve Fibonacci numbers. They form four circles that cut the x-axis at points…
The concept of graph compositions is related to several number theoretic concepts, including partitions of positive integers and the cardinality of the power set of finite sets. This paper examines graph compositions where the total number…
Fractals and quasiperiodic structures share self-similarity as a structural property. Motivated by the link between Fibonacci fractals and quasicrystals which are scaled by the golden mean ratio $\frac{1+\sqrt{5}}{2}$, we introduce and…
Given a set of points in the plane, we show that the $\theta$-graph with 5 cones is a geometric spanner with spanning ratio at most $\sqrt{50 + 22 \sqrt{5}} \approx 9.960$. This is the first constant upper bound on the spanning ratio of…
By using definition of Golden derivative, corresponding Golden exponential function and Fibonomial coefficients, we introduce generating functions for Bernoulli-Fibonacci polynomials and related numbers. Properties of these polynomials and…
There are useful and useless golden ratios. The useful one helps in traffic. The useless and rather mysterious one arises in hydrodynamics of point vortices, which we discuss in detail.
The domatic number of a graph is the maximum number of vertex disjoint dominating sets that partition the vertex set of the graph. In this paper we consider the fractional variant of this notion. Graphs with fractional domatic number 1 are…
A golden-ratio-based rectangular tiling of the first quadrant of the Euclidean plane is constructed by drawing vertical and horizontal grid lines which are located at all even powers of $\phi$ along one axis, and at all odd powers of $\phi$…
We study the linear fractional transformations in the Hecke group $G(\Phi)$ where $\Phi$ is either root of $x^2 - x -1$ (the larger root being the "golden ratio" $\phi = 2 \cos \frac {\pi}5$.) Let $g \in G(\Phi)$ and let $z$ be a generic…
As is well-known, the ratio of adjacent Fibonacci numbers tends to phi = (1 + sqrt(5))/2, and the ratio of adjacent Tribonacci numbers (where each term is the sum of the three preceding numbers) tends to the real root eta of X^3 - X^2 - X -…
There is a family of vector bundles over the moduli space of stable curves that, while first appearing in theoretical physics, has been an active topic of study for algebraic geometers since the 1990s. By computing the rank of the…
We give an asymptotic estimate of the number of numerical semigroups of a given genus. In particular, if $n_g$ is the number of numerical semigroups of genus $g$, we prove that $n_g$ tends to $S \phi^g$, where $\phi$ is the golden ratio,…
We provide a family of expressions of $\pi$ in terms of the golden ratio $\phi$ in the same spirit of the formula obtained by Bailey, Borwein and Plouffe for $\pi$. Connection with cyclotomic polynomials is outlined.
In this thesis we examined mathematical properties of Fibonacci numbers and applications of this numbers in the nature,geometry and economy.We obtained Golden section and proved some mathematical identities using Golden section. Infinity of…
For a given number of colors, $s$, the guessing number of a graph is the (base $s$) logarithm of the cardinality of the largest family of colorings of the vertex set of the graph such that the color of each vertex can be determined from the…
The Hall ratio of a graph G is the maximum of |V(H)|/alpha(H) over all subgraphs H of G. Clearly, the Hall ratio of a graph is a lower bound for the fractional chromatic number. It has been asked whether conversely, the fractional chromatic…