Replacement product

From HandWiki

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.[1] 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 (ij) in E(H), the vertex (ui) is adjacent to (uj) in R. Furthermore, for each edge (uv) in E(G), if v is the ith neighbor of u and u is the jth neighbor of v, the vertex (ui) is adjacent to (vj) in R.

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

References

  1. Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (7 August 2006). "Expander graphs and their applications". Bulletin of the American Mathematical Society 43 (4): 439–562. doi:10.1090/S0273-0979-06-01126-8. 

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