Counting Cliques in Finite Distant Graphs

TL;DR

This paper develops formulas for counting cliques in distant graphs of projective lines over finite rings, including a decomposition theorem based on ring structure, advancing understanding of graph properties in algebraic settings.

Contribution

It introduces new counting formulas and a decomposition theorem for distant graphs over finite rings, linking graph structure to ring decomposition.

Findings

Derived counting formulas for cliques in distant graphs.

Proved a decomposition theorem relating graph structure to ring decomposition.

Enhanced understanding of the algebraic structure of these graphs.

Abstract

We state and prove some counting formulas relating to cliques in the distant graphs of projective lines over finite rings. As a preliminary to this, we prove a decomposition theorem for the graphs in terms of the direct-product decomposition of their rings.