The number of cyclic configurations of type $(v_3)$ and the isomorphism problem

TL;DR

This paper derives a formula for counting non-isomorphic connected cyclic configurations of type (v_3) and proves a Bays-Lambossy type theorem for configurations with specific point counts, advancing combinatorial classification methods.

Contribution

It provides a closed-form formula for the number of such configurations and extends classification results to cases where the number of points is a prime power or product of two primes.

Findings

Derived a closed formula for non-isomorphic connected cyclic configurations of type (v_3)

Proved a Bays-Lambossy type theorem for configurations with prime power or product of two primes points

Enhanced understanding of automorphism structures in cyclic configurations

Abstract

A configuration of points and lines is cyclic if it has an automorphism which permutes its points in a full cycle. A closed formula is derived for the number of non-isomorphic connected cyclic configurations of type (v_3), i.e., which have v points and lines, and each point/line is incident with exactly 3 lines/points. In addition, a Bays-Lambossy type theorem is proved for cyclic configurations if the number of points is a product of two primes or a prime power.