Proof of a conjecture on monomial graphs

Let e be a positive integer, p be an odd prime, $q=p^e$, and ${\mathbb{F}}_q$ be the finite field of q elements. Let $f,g\in {\mathbb{F}}_q[X,Y]$. The graph $G_q(f,g)$ is a bipartite graph with vertex partitions $P={\mathbb{F}}_q^3$ and $L={\mathbb{F}}_q^3$, and edges defined as follows: a vertex $(p)=(p_1,p_2,p_3)\in P$ is adjacent to a vertex $[l]=[l_1,l_2,l_3]\in L$ if and only if $p_2+l_2=f(p_1,l_1)$ and $p_3+l_3=g(p_1,l_1)$. If $f=XY$ and $g=X{Y}^2$, the graph $G_q(XY,X{Y}^2)$ contains no cycles of length less than eight and is edge-transitive. Motivated by certain questions in extremal graph theory and finite geometry, people search for examples of graphs $G_q(f,g)$ containing no cycles of length less than eight and not isomorphic to the graph $G_q(XY,X{Y}^2)$, even without requiring them to be edge-transitive. So far, no such graphs $G_q(f,g)$ have been found. It was conjectured that if both f and g are monomials, then no such graphs exist. In this paper we prove the conjecture.     

Counting numerical semigroups by genus and even gaps

Let $n_g$ be the number of numerical semigroups of genus $g$. We present an approach to compute $n_g$ by using even gaps, and the question: Is it true that $n_{g+1}>n_g$? is investigated. Let $N_{\gamma }\left(g\right)$ be the number of numerical semigroups of genus $g$ whose number of even gaps equals $\gamma$. We show that $N_{\gamma }\left(g\right)=N_{\gamma }\left(3\gamma \right)$ for $\gamma \le ?g∕3?$ and $N_{\gamma }\left(g\right)=0$ for $\gamma >?2g∕3?$; thus the question above is true provided that $N_{\gamma }(g+1)>N_{\gamma }\left(g\right)$ for $\gamma =?g∕3?+1,\dots ,?2g∕3?$. We also show that $N_{\gamma }\left(3\gamma \right)$ coincides with $f_{\gamma }$, the number introduced by Bras-Amorós (2012) in connection with semigroup-closed sets. Finally, the stronger possibility $f_{\gamma }～{\phi }^{2\gamma }$ arises being $\phi =(1+\sqrt{5})∕2$ the golden number.     

On the difficulty of finding spines

We prove that the set of symplectic lattices in the Siegel space ${\mathfrak{h}}_g$ whose systoles generate a subspace of dimension at least 3 in ${\mathbb{R}}^{2g}$ does not contain any $\mathrm{Sp}(2g,\mathbb{Z})$-equivariant deformation retract of ${\mathfrak{h}}_g$.     

Compact simple Lie groups admitting left-invariant Einstein metrics that are not geodesic orbit

In this article, we prove that the compact simple Lie groups $\mathrm{S}\mathrm{U}(n)$ for $n\ge 6$, $\mathrm{S}\mathrm{O}(n)$ for $n\ge 7$, $\mathrm{S}\mathrm{p}(n)$ for $n\ge 3$, ${\mathrm{E}}_6,{\mathrm{E}}_7,{\mathrm{E}}_8$, and ${\mathrm{F}}_4$ admit left-invariant Einstein metrics that are not geodesic orbit. This gives a positive answer to an open problem recently posed by Nikonorov.