A characterization of graphs with regular distance-2 graphs

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