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Sum Index, Difference Index and Exclusive Sum Number of Graphs.

John Haslegrave1

  • 1Mathematical Institute, University of Oxford, Oxford, UK.

Graphs and Combinatorics
|June 29, 2026
PubMed
Summary

This study investigates graph invariants, specifically the sum index and difference index. We demonstrate that a proposed exact relationship between these graph properties is not generally true, disproving a recent conjecture.

Keywords:
Difference indexGraph labellingSum graphsSum index

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Area of Science:

  • Graph theory
  • Combinatorics
  • Discrete mathematics

Background:

  • Recent conjectures by Harrington et al. proposed relationships between graph invariants: sum index, difference index, and exclusive sum number.
  • These conjectures explored potential equalities and inequalities among these graph properties.

Purpose of the Study:

  • To rigorously evaluate two recent conjectures regarding graph invariants.
  • To determine the validity of the proposed relationships between the sum index, difference index, and exclusive sum number.
  • To provide new bounds and explicit constructions related to these graph parameters.

Main Methods:

  • Theoretical analysis of graph properties.
  • Development of counterexamples to disprove conjectured relationships.
  • Construction of explicit graph examples to demonstrate theoretical findings.
  • Establishment of new bounds for the sum and difference indices.

Main Results:

  • The conjecture proposing an exact relationship between the sum index and difference index is disproven; the discrepancy can be arbitrarily large.
  • New upper and lower bounds for the sum index and difference index are established.
  • The conjecture that the exclusive sum number can arbitrarily exceed the sum index is confirmed through an explicit construction.

Conclusions:

  • The initial conjecture on the exact relationship between sum and difference indices is false.
  • The modified inequality conjecture regarding sum and difference indices is also shown to be generally false.
  • An explicit construction validates the conjecture on the arbitrary difference between exclusive sum number and sum index.