Intersection density of cubic symmetric graphs

Abstract

Two elements g, h of a permutation group G acting on a set V are said to be intersecting if g(v) = h(v) for some v in V. More generally, a subset mathcal {F} of G is an intersecting set if every pair of elements of mathcal {F} is intersecting. The intersection density rho (G) of a transitive permutation group G is the maximum value of the quotient |mathcal {F}|/|G_v| where mathcal {F} runs over all intersecting sets in G and G_v is the stabilizer of v in V. A vertex-transitive graph X is intersection density stable if any two transitive subgroups of {textrm{Aut}},(X) have the same intersection density. This paper studies the above concepts in the context of cubic symmetric graphs. While a 1-regular cubic symmetric graph is necessarily intersection density stable, the situation for 2-arc-regular cubic symmetric graphs is more complex. A necessary condition for a 2-arc-regular cubic symmetric graph admitting a 1-arc-regular subgroup of automorphisms to be intersection density stable is given, and an infinite family of such graphs is constructed.