Replacement product

In graph theory, the replacement product of two graphs is a graph product that can be used to reduce the degree of a graph while maintaining its connectivity.

Suppose $G$ is a $d$-regular graph and $H$ is an $e$-regular graph with vertex set ${0, …, d – 1}.$ Let $R$ denote the replacement product of $G$ and $H$. The vertex set of $R$ is the Cartesian product $V(G) × V(H)$. For each vertex $u$ in $V(G)$ and for each edge $(i, j)$ in $E(H)$, the vertex $(u, i)$ is adjacent to $(u, j)$ in $R$. Furthermore, for each edge $(u, v)$ in $E(G)$, if $v$ is the $i$th neighbor of $u$ and $u$ is the $j$th neighbor of $v$, the vertex $(u, i)$ is adjacent to $(v, j)$ in $R$.

If $H$ is an $e$-regular graph, then $R$ is an $(e + 1)$-regular graph.