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On the Wiener-like root-indices of graphs

Simon Brezovnik, Matthias Dehmer, Niko Tratnik, Petra Žigert Pleteršek

Abstract

In this paper, we examine roots of graph polynomials where those roots can be considered as structural graph measures. More precisely, we prove analytical results for the roots of certain modified graph polynomials and also discuss numerical results. As polynomials, we use, e.g., the Hosoya, the Schultz, and the Gutman polynomial which belong to an interesting family of degree-distance-based graph polynomials; they constitute so-called counting polynomials with non-negative integers as coefficients and the roots of their modified versions have been used to characterize the topology of graphs. Our results can be applied for the quantitative characterization of graphs. Besides analytical results, we also investigate other properties of those measures such as their degeneracy which is an undesired aspect of graph measures. It turns out that the measures representing roots of graph polynomials possess high discrimination power on exhaustively generated trees, which outperforms standard versions of these indices. Furthermore, a new measure is introduced that allows us to compare different topological indices in terms of structure sensitivity and abruptness.

On the Wiener-like root-indices of graphs

Abstract

In this paper, we examine roots of graph polynomials where those roots can be considered as structural graph measures. More precisely, we prove analytical results for the roots of certain modified graph polynomials and also discuss numerical results. As polynomials, we use, e.g., the Hosoya, the Schultz, and the Gutman polynomial which belong to an interesting family of degree-distance-based graph polynomials; they constitute so-called counting polynomials with non-negative integers as coefficients and the roots of their modified versions have been used to characterize the topology of graphs. Our results can be applied for the quantitative characterization of graphs. Besides analytical results, we also investigate other properties of those measures such as their degeneracy which is an undesired aspect of graph measures. It turns out that the measures representing roots of graph polynomials possess high discrimination power on exhaustively generated trees, which outperforms standard versions of these indices. Furthermore, a new measure is introduced that allows us to compare different topological indices in terms of structure sensitivity and abruptness.

Paper Structure

This paper contains 6 sections, 12 theorems, 42 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Explicit formulas for some degree-distance-based polynomials
  4. Roots of modified degree-distance-based polynomials
  5. Numerical results
  6. Summary and conclusion

Key Result

Proposition 1

For the complete graph $K_n$ with $n$ vertices, $n \geq 1$, it holds

Figures (1)

  • Figure 1: Chemical graph $G$ of phenanthrene.

Theorems & Definitions (17)

  • Proposition 1
  • Proposition 2
  • proof
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Lemma 7
  • Proposition 8
  • proof
  • ...and 7 more