Two elements ▫$g, h$▫ of a permutation group ▫$G$▫ acting on a set ▫$V$▫ are said to be {\em intersecting} if ▫$g(v) = h(v)$ ▫ for some ▫$v \in V$▫. More generally, a subset ▫$\mathcal{F}$▫ of ▫$G$▫ ...is an {\em 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 {\em intersection density stable} if any two transitive subgroups of ▫$\Aut(X)$▫ 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.
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