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Dot Product Representations of Graphs Using Tropical Arithmetic

Sean Bailey, David Brown, Michael Snyder, Nicole Turner

Abstract

A dot-product representation of a graph is a mapping of its vertices to vectors of length $k$ so that vertices are adjacent if and only if the inner product (a.k.a. dot product) of their corresponding vertices exceeds some threshold. Minimizing dimension of the vector space into which the vectors must be mapped is a typical focus. We investigate this and structural characterizations of graphs whose dot product representations are mappings into the tropical semi-rings of min-plus and max-plus. We also observe that the minimum dimension required to represent a graph using a \emph{tropical representation} is equal to the better-known threshold dimension of the graph; that is, the minimum number of subgraphs that are threshold graphs whose union is the graph being represented.

Dot Product Representations of Graphs Using Tropical Arithmetic

Abstract

A dot-product representation of a graph is a mapping of its vertices to vectors of length so that vertices are adjacent if and only if the inner product (a.k.a. dot product) of their corresponding vertices exceeds some threshold. Minimizing dimension of the vector space into which the vectors must be mapped is a typical focus. We investigate this and structural characterizations of graphs whose dot product representations are mappings into the tropical semi-rings of min-plus and max-plus. We also observe that the minimum dimension required to represent a graph using a \emph{tropical representation} is equal to the better-known threshold dimension of the graph; that is, the minimum number of subgraphs that are threshold graphs whose union is the graph being represented.
Paper Structure (13 sections, 26 theorems, 17 equations, 5 figures)

This paper contains 13 sections, 26 theorems, 17 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic Definitions
  4. Graph Classes
  5. The Dot-Product Representation of a Graph
  6. Tropical Arithmetic
  7. Results
  8. Results with $\Theta (G)$
  9. Other Results
  10. Applications
  11. A Min-Plus Tropical Dot Product Graph Application
  12. A Max-Plus Tropical Dot Product Graph Application
  13. Conclusion

Key Result

Theorem 2.1

(Chvatal) For every graph $G$ on $n$ vertices we have $\Theta(G)\le n-\alpha(G)$. Furthermore, if $G$ is triangle-free, then $\Theta(G) = n - \alpha(G)$.

Figures (5)

  • Figure 2: A DPR of a Graph
  • Figure 3: A caterpillar with its vertices labeled
  • Figure 4: A min-plus $2$-dot product representation of $B$
  • Figure 5: Tropical Dot Product Graph Generated For Students
  • Figure 6: Tropical Dot Product Graph Generated For Mutual Funds

Theorems & Definitions (49)

  • Theorem 2.1
  • Proposition 3.0.1
  • Theorem 5.1
  • proof
  • Corollary 5.1.1
  • proof
  • Corollary 5.1.2
  • proof
  • Theorem 5.2
  • proof
  • ...and 39 more