Degrees in Link Graphs of Regular Graphs

We analyse an extremal question on the degrees of link graphs a finite regular graph, that is, subgraphs induced by non-trivial spheres. show if $G$ is $d$-regular and connected but not complete then some graph has minimum degree at most $\lfloor 2d/3\rfloor-1$, sufficiently large in terms $d$ d/2\rfloor-1$; both bounds are best possible. also give corresponding best-possible result for problem where balls, rather than spheres, considered. motivate these questions posing conjecture concerning expansion bounded-degree graphs, together with heuristic justification thereof.