Let \( G \) be a finite group. The directed power graph \( \vec{\mathcal{P}}(G) \) of \( G \) takes \( G \) as its vertex set with an edge from each element to each of its powers (other than itself). The (undirected) power graph \( \mathcal{P}(G) \) of \( G \) ignores orientation.

The directed power graph of \(Z_6 \)

In [GroupIneq] and [EdgeMax], we show that the power graph of the cyclic group has the maximum number of edges and bidirected edges among finite groups of any given order. These results are equivalent to the following inequalities for any group \(G\) of order \(n\): \[ \sum_{d|n} (2d - \phi(d))\phi(d) \geq \sum_{g\in G} 2o(g) - \phi(o(g)), \] \[ \sum_{d|n} \phi(d)^2 \geq \sum_{g\in G} \phi(o(g)). \] In [EulerIneq] we consider the size of the maximum clique in power graphs of the cyclic group of order \(n\). It is given by the function \(\chi(n)\) satisfying \(\chi(1) = 1\) and \(\chi(n) =\phi(n) + \chi(n/p)\), where \(p\) is the least prime divisor of \(n\).

For all integers \( i \), let \( F_i \) and \( L_i \) denote the \( i^{th} \) Fibonacci and Lucas numbers. The \(i^{th} \) Fibonacci and Lucas vectors of length \( d \) consist of \( d \) consecutive Fibonaci or Lucas numbers starting at the \(i^{th}\): \[ \vec{f}_i^d = ( F_i \ F_{i+1} \ \cdots \ F_{i+d}), \quad \vec{l}_i^d = ( L_i \ L_{i+1} \ \cdots \ L_{i+d} ). \] Let \( \alpha = \frac{1+\sqrt{5}}{2}\) and \( \beta = \frac{1-\sqrt{5}}{2}\), and write \[ \vec{a}^d = ( 1 \ \alpha \ \cdots \ \alpha^{d-1}), \quad \vec{b}^d = ( 1 \ \beta \ \cdots \ \beta^{d-1} ). \] The Binet formulas are \[ \vec{f}_i^d = \frac{1}{\alpha-\beta}( \alpha^i \vec{a}^d - \beta^i \vec{b}^d), \quad \vec{l}_i^d = \alpha^i \vec{a}^d + \beta^i \vec{b}^d.\]

In [FibForm], we derive some identities by taking dot products of Fibonacci and Lucas vectors (and their reverses) and simplifying the Binet formulas using the formula for a finite geometric sum.

In [LucHyp], we describe some remarkable geometric properties of Fibonacci and Lucas vectors using the fact that the Binet formulas imply that Fibonacci and Lucas vectors lie on hyperbolas with \( \vec{a}^d \) and \( \vec{b}^d \) as asymptotes. This generalizes the known result for \( d=2 \).