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Note on extremal problems about connected subgraph sums.

Stijn Cambie1, Carla Groenland2

  • 1Department of Computer Science, KU Leuven Campus Kulak-Kortrijk, 8500 Kortrijk, Belgium.

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Summary
This summary is machine-generated.

This study introduces a novel vertex assignment method for graphs, ensuring unique connected subgraph sums for distinct graph structures. This resolves a long-standing problem in graph theory, enhancing graph identification capabilities.

Keywords:
Connected subgraph sumsGolomb rulerGraph labellingGraph reconstructionSidon set

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Area of Science:

  • Graph Theory
  • Combinatorics
  • Discrete Mathematics

Background:

  • Graphs are fundamental structures in mathematics and computer science.
  • Connected subgraph sums are derived from vertex assignments on graphs.
  • Distinguishing graphs based on these sums is a key theoretical challenge.

Purpose of the Study:

  • To resolve a problem posed by O.-H. S. Lo regarding unique connected subgraph sums.
  • To establish a vertex assignment strategy that guarantees distinct subgraph sum sets for non-isomorphic graphs.
  • To explore properties of vertex assignments that yield unique connected subgraph sums.

Main Methods:

  • Defined connected subgraph sum for a graph G with vertex assignment c.
  • Constructed a specific vertex assignment c for an n-vertex graph G.
  • Proved that this assignment ensures S(G, c) ≠ S(G', c') for non-isomorphic graphs G' and assignments c'.

Main Results:

  • Demonstrated the existence of a vertex assignment c: V(G) → {1, ..., 12n^2} for any n-vertex graph G.
  • Showed that this assignment guarantees unique collections of connected subgraph sums for non-isomorphic graphs.
  • Provided insights into vertex assignments that produce distinct connected subgraph sums.

Conclusions:

  • The study successfully provides a method to uniquely identify graphs using connected subgraph sums.
  • This resolves a significant problem in graph theory by establishing a strong form of graph distinctness.
  • The findings have implications for graph isomorphism testing and structural graph analysis.