Strong product of graphs

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The king's graph, a strong product of two path 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' .

It is the union of the Cartesian product and the tensor product.

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 been used to denote the tensor product of graphs.[4]

See also[]

References[]

  1. ^ Imrich, Wilfried; Klavžar, Sandi; Rall, Douglas F. (2008), Graphs and their Cartesian Product, A. K. Peters, ISBN 978-1-56881-429-2.
  2. ^ Sabidussi, G. (1960). "Graph multiplication". Math. Z. 72: 446–457. doi:10.1007/BF01162967. hdl:10338.dmlcz/102459. MR 0209177.
  3. ^ 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.
  4. ^ See page 2 of Lovász, László (1979), "On the Shannon Capacity of a Graph", IEEE Transactions on Information Theory, IT-25 (1): 1–7, doi:10.1109/TIT.1979.1055985.
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