A characterization of graphs with regular distance-2 graphs

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≥5 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≤8. 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.