Strong product of graphs
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
- any two distinct vertices (u,u') and (v,v') are adjacent in G ⊠ H if and only if:
- u is adjacent to v and u'=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 and AND product. It was first introduced by Sabidussi in 1960.[2]
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]
Shannon capacity and Lovász number
The Shannon capacity of a graph is defined from the independence number of its strong products with itself, by the formula
![\Theta (G)=\sup _{k}{\sqrt[ {k}]{\alpha (G^{k})}}=\lim _{{k\rightarrow \infty }}{\sqrt[ {k}]{\alpha (G^{k})}},](../I/m/95dbf3f6d9ee2bf131f3c9cf9c15410d5f22c720.svg)
László Lovász showed that Lovász theta function is multiplicative:[4]
He used this fact to upper bound the Shannon capacity by the Lovász number.
See also
References
- ↑ Imrich, Wilfried; Klavžar, Sandi; Rall, Douglas F. (2008), Graphs and their Cartesian Products, 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 Lemma 2 and Theorem 7 in Lovász, László (1979), "On the Shannon Capacity of a Graph", IEEE Transactions on Information Theory, IT-25 (1), doi:10.1109/TIT.1979.1055985.
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