A proof of the cyclotomic conjecture and the non-existence of almost Moore digraphs

For $n>2$ and $k>1$, define the polynomial $$F_{n,k}(x)={\mathrm{\Phi }}_n(1+x+\cdots +x^k),$$ where ${\mathrm{\Phi }}_n$ denotes the $n$-th cyclotomic polynomial. The \emph{cyclotomic conjecture} proposed by Gimbert (1999) exactly describes the irreducibility of $F_{n,k}(x)$ over $\mathbb{Q}$ in terms of $n$ and $k$. Conde, Gimbert, Gonz'{a}lez, Miller and Miret (2014) established that the cyclotomic conjecture, if true, would imply the non-existence of almost Moore digraphs - a well-known open question concerning the directed degree-diameter problem. In this article, we prove the cyclotomic conjecture and, as a consequence, show that there are no almost Moore digraphs with maximum out-degree $d$ and diameter $k$ for any $d>1$ and $k>2$.