Strong product of graphs
In graph theory, the strong product G ⊠ H of graphs G and H is a graph such that[1]
- the vertex set of G ⊠ H is the Cartesian product V(G) × V(H); and
- distinct vertices (u,u' ) and (v,v' ) are adjacent in G ⊠ H if and only if:
- u = v and u' is adjacent to v' , or
- u' = v' and u is adjacent to v, or
- u is adjacent to v and u' is adjacent to v'
The strong product is also called the normal product or the AND product[citation needed]. It was first introduced by Sabidussi in 1960.[2] In that setting, the strong product is contrasted against a weak product, but the two are different only when applied to infinitely many factors.
For example, the king's graph, a graph whose vertices are squares of a chessboard and whose edges represent possible moves of a chess king, is a strong product of two path graphs.[3]
Care should be exercised when encountering the term strong product in the literature, since it has also[4] been used to denote the tensor product of graphs.
See also
References
- ^ Imrich, Wilfried; Klavžar, Sandi; Rall, Douglas F. (2008), Graphs and their Cartesian Product, A. K. Peters, ISBN 1-56881-429-1.
- ^ Sabidussi, G. (1960). "Graph multiplication". Math. Z. 72: 446–457. doi:10.1007/BF01162967. MR 0209177.
- ^ Berend, Daniel; Korach, Ephraim; Zucker, Shira (2005), "Two-anticoloring of planar and related graphs" (PDF), 2005 International Conference on Analysis of Algorithms, Discrete Mathematics & Theoretical Computer Science Proceedings, Nancy: Association for Discrete Mathematics & Theoretical Computer Science, pp. 335–341, MR 2193130.
- ^ See page 2 of Lovász, László (1979), "On the Shannon Capacity of a Graph", IEEE Transactions on Information Theory, IT-25 (1), doi:10.1109/TIT.1979.1055985.