Limits and Periodicity of Metamour $2$-Distance Graphs

TL;DR

This paper investigates the iterative behavior of 2-distance graphs, classifies graphs with specific periodicities, and analyzes particular graph families to understand their eventual periodicity and limit structures.

Contribution

It classifies connected graphs with period 3, partially characterizes those with period 2, and studies the eventual periodicity of generalized Petersen graphs and complete m-ary trees.

Findings

Classified connected graphs with period 3.

Partially characterized graphs with period 2.

Established eventual period as 2 for Petersen and m-ary trees.

Abstract

Given a finite simple graph $G$, let $\operatorname{M}(G)$ denote its 2-distance graph, in which two vertices are adjacent if and only if they have distance 2 in $G$. In this paper, we consider the periodic behavior of the sequence $G, \operatorname{M}(G), \operatorname{M}^2(G), \operatorname{M}^3(G), \ldots$ obtained by iterating the 2-distance operation. In particular, we classify the connected graphs with period 3, and we partially characterize those with period 2. We then study two families of graphs whose 2-distance sequence is eventually periodic: namely, generalized Petersen graphs and complete $m$-ary trees. For each family, we show that the eventual period is 2, and we determine the pre-period and the two limit graphs of the sequence.