Families of Small Regular Graphs of Girth 5

TL;DR

This paper constructs new small regular graphs with girth 5 for specific prime degrees by applying reduction and amalgamation techniques to Levi graphs of elliptic semiplanes, improving known vertex counts.

Contribution

It introduces a novel method of using reductions and amalgams on Levi graphs to produce smaller regular graphs with girth 5 for various prime degrees.

Findings

Constructed $(q+3)$-regular graphs of girth 5 for q=13,17,19, and all primes ≥23.

Produced a 13-regular graph of girth 5 with 236 vertices from $B_{11}$.

Achieved graphs with fewer vertices than previously known for these degrees.

Abstract

In this paper we obtain $(q+3)$--regular graphs of girth 5 with fewer vertices than previously known ones for $q=13,17,19$ and for any prime $q \ge 23$ performing operations of reductions and amalgams on the Levi graph $B_q$ of an elliptic semiplane of type ${\cal C}$. We also obtain a 13-regular graph of girth 5 on 236 vertices from $B_{11}$ using the same technique.