Geodesic graphs for geodesic orbit Finsler $(\alpha,\beta)$ metrics on spheres

Invariant geodesic orbit Finsler $(\alpha,\beta)$ metrics $F$ which arise from Riemannian geodesic orbit metrics $\alpha$ on spheres are determined. The relation of Riemannian geodesic graphs with Finslerian geodesic graphs proved in a previous work is now illustrated with explicit constructions. Interesting examples are found such that $(G/H,\alpha)$ is Riemannian geodesic orbit space, but for the geodesic orbit property of $(G/H,F)$ the isometry group has to be extended. It is also shown that projective spaces other than ${\mathbb{R}}P^n$ do not admit invariant purely Finsler $(\alpha,\beta)$ metrics.