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On Sum Graphs over Some Magmas

António Machiavelo, Rogério Reis

Abstract

We consider the notions of sum graph and of relaxed sum graph over a magma, give several examples and results of these families of graphs over some natural magmas. We classify the cycles that are sum graphs for the magma of the subsets of a set with the operation of union, determine the abelian groups that provide a sum labelling of $C_4$, and show that $C_{4\ell}$ is a sum graph over the abelian group $\mathbb{Z}_f\times\mathbb{Z}_f$, where $f=f_{2\ell}$ is the corresponding Fibonacci number. For integral sum graphs, we give a linear upper bound for the radius of matchings, improving Harary's labelling for this family of graphs, and give the exact radius for the family of totally disconnected graphs. We found integer labellings for the 4D-cube, giving a negative answer to a question of Melnikov and Pyatikin, actually showing that the 4D-cube has infinitely many primitive labellings. We have also obtained some new results on mod sum graphs and relaxed sum graphs. Finally, we show that the direct product operation is closed for strong integral sum graphs.

On Sum Graphs over Some Magmas

Abstract

We consider the notions of sum graph and of relaxed sum graph over a magma, give several examples and results of these families of graphs over some natural magmas. We classify the cycles that are sum graphs for the magma of the subsets of a set with the operation of union, determine the abelian groups that provide a sum labelling of , and show that is a sum graph over the abelian group , where is the corresponding Fibonacci number. For integral sum graphs, we give a linear upper bound for the radius of matchings, improving Harary's labelling for this family of graphs, and give the exact radius for the family of totally disconnected graphs. We found integer labellings for the 4D-cube, giving a negative answer to a question of Melnikov and Pyatikin, actually showing that the 4D-cube has infinitely many primitive labellings. We have also obtained some new results on mod sum graphs and relaxed sum graphs. Finally, we show that the direct product operation is closed for strong integral sum graphs.
Paper Structure (22 sections, 15 theorems, 39 equations)

This paper contains 22 sections, 15 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Sum Graphs over Magmas
  3. Magmas on Sets
  4. The Union and the Intersection
  5. The Symmetric Difference
  6. The Complements of the Intersection and of the Union
  7. The $C_4$ Problem on Abelian Groups
  8. The $C_n$ Problem
  9. A Serendipitious Conjecture
  10. More Examples
  11. A $C_9$ Example
  12. $C_{4\ell}$ as a Sum Graph over an Abelian Group
  13. Integral Sum Graphs
  14. The radius of a $\mathbb{Z}-$graph.
  15. The Z3 Theorem Prover and Presburger Arithmetic
  16. ...and 7 more sections

Key Result

Proposition 3.1

When $n\geq 5$ is an odd integer, the graph $C_n$ is not a sum graph over any magma of subsets of a set with the union operation. The graph $C_{2k}$ is a sum graph over that magma for a set with $2k$ elements.

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 4.1
  • Proposition 5.1
  • Proposition 5.2
  • Theorem 8.1
  • ...and 15 more