On regular induced subgraphs of generalized polygons

The cage problem asks for the smallest number $c(k,g)$ of vertices in a $k$-regular graph of girth $g$ and graphs meeting this bound are known as cages. While cages are known to exist for all integers $k \ge 2$ and $g \ge 3$, the exact value of $c(k, g)$ is known only for some small values of $k, g$ and three infinite families where $g \in \{6, 8, 12\}$ and $k - 1$ is a prime power. These infinite families come from the incidence graphs of generalized polygons. Some of the best known upper bounds on $c(k,g)$ for $g \in \{6, 8, 12\}$ have been obtained by constructing small regular induced subgraphs of these cages. In this paper, we first use the Expander Mixing Lemma to give a general lower bound on the size of an induced $k$-regular subgraph of a regular bipartite graph in terms of the second largest eigenvalue of the host graph. We use this bound to show that the known construction of $(k,6)$-graphs using Baer subplanes of the Desarguesian projective plane is the best possible. For generalized quadrangles and hexagons, our bounds are new. In particular, we improve the known lower bound on the size of a $q$-regular induced subgraphs of the classical generalized quadrangle $\mathsf{Q}(4,q)$ and show that the known constructions are asymptotically sharp. For prime powers $q$, we also improve the known upper bounds on $c(q,8)$ and $c(q,12)$ by giving new geometric constructions of $q$-regular induced subgraphs in the symplectic generalized quadrangle $\mathsf{W}(3,q)$ and the split Cayley hexagon $\mathsf{H}(q)$, respectively. Our constructions show that $$c(q,8) \le 2(q^3 - q\sqrt{q} - q)$$ for $q$ an even power of a prime, and $$c(q, 12) \le 2(q^5 - 3q^3)$$ for all prime powers $q$. For $q \in \{3,4,5\}$ we also give a computer classification of all $q$-regular induced subgraphs of the classical generalized quadrangles of order $q$.