Dot product representation of a graph

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A dot product representation of a simple graph is a method of representing a graph using vector spaces and the dot product from linear algebra. Every graph has a dot product representation. [1] [2] [3]

Contents

Definition

Let G be a graph with vertex set V. Let F be a field, and f a function from V to Fk such that xy is an edge of G if and only if f(x)·f(y) ≥ t. This is the dot product representation of G. The number t is called the dot product threshold, and the smallest possible value of k is called the dot product dimension. [1]

Properties

See also

References

  1. 1 2 3 4 Fiduccia, Charles M.; Scheinerman, Edward R.; Trenk, Ann; Zito, Jennifer S. (1998), "Dot product representations of graphs", Discrete Mathematics, 181 (1–3): 113–138, doi: 10.1016/S0012-365X(97)00049-6 , MR   1600755 .
  2. ↑ Reiterman, J.; Rödl, V.; Šiňajová, E. (1989), "Embeddings of graphs in Euclidean spaces", Discrete & Computational Geometry, 4 (4): 349–364, doi: 10.1007/BF02187736 , MR   0996768 .
  3. ↑ Reiterman, J.; Rödl, V.; Šiňajová, E. (1992), "On embedding of graphs into Euclidean spaces of small dimension", Journal of Combinatorial Theory, Series B, 56 (1): 1–8, doi: 10.1016/0095-8956(92)90002-F , MR   1182453 .
  4. ↑ Kang, Ross J.; Lovász, László; Müller, Tobias; Scheinerman, Edward R. (2011), "Dot product representations of planar graphs", Electronic Journal of Combinatorics, 18 (1): Paper 216, doi: 10.37236/703 , MR   2853073 .
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