A characterization of graphs with regular distance-$2$ graphs

TL;DR

This paper characterizes graphs where each vertex has a fixed number of vertices at distance two, providing complete classifications for cases where this number is 0, 1, or 2, and exploring related counting sequences.

Contribution

It offers a complete characterization of $k$-metamour-regular graphs for $k=0,1,2$, including constructions, exceptions, and enumeration insights.

Findings

Characterization of 2-metamour-regular graphs as joins of complements of cycles, cycles, or exceptional graphs.

Complete classification of graphs with at most one metamour per vertex.

Analysis of counting sequences for unlabeled graphs in these classes.

Abstract

For non-negative integers~$k$, we consider graphs in which every vertex has exactly $k$ vertices at distance~$2$, i.e., graphs whose distance-$2$ graphs are $k$-regular. We call such graphs $k$-metamour-regular motivated by the terminology in polyamory. While constructing $k$-metamour-regular graphs is relatively easy -- we provide a generic construction for arbitrary~$k$ -- finding all such graphs is much more challenging. We show that only $k$-metamour-regular graphs with a certain property cannot be built with this construction. Moreover, we derive a complete characterization of $k$-metamour-regular graphs for each $k=0$, $k=1$ and $k=2$. In particular, a connected graph with~$n$ vertices is $2$-metamour-regular if and only if $n\ge5$ and the graph is a join of complements of cycles (equivalently every vertex has degree~$n-3$), a cycle, or one of $17$ exceptional graphs with $n\le8$. Moreover, a characterization of graphs in which every vertex has at most one metamour is acquired. Each characterization is accompanied by an investigation of the corresponding counting sequence of unlabeled graphs.