Degrees in link graphs of regular graphs

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