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On the Second-Order Wiener Ratios of Iterated Line Graphs

Mohammad Ghebleh, Ali Kanso

Abstract

The Wiener index W(G) of a graph G is the sum of distances between all unordered pairs of its vertices. Dobrynin and Mel'nikov [in: Distance in Molecular Graphs - Theory, 2012, p. 85-121] propose the study of estimates for extremal values of the ratio R_k(G) = W(L^k(G))/W(G) where L^k(G) denotes the k-th iterated line graph of G. Hriňáková, Knor and Škrekovski [Art Discrete Appl. Math. 1 (2018) #P1.09] prove that for each k>2, the path P_n has the smallest value of the ratio R_k among all trees of large order n, and they conjecture that the same holds for the case k=2. We give a counterexample of every order n>21 to this conjecture.

On the Second-Order Wiener Ratios of Iterated Line Graphs

Abstract

The Wiener index W(G) of a graph G is the sum of distances between all unordered pairs of its vertices. Dobrynin and Mel'nikov [in: Distance in Molecular Graphs - Theory, 2012, p. 85-121] propose the study of estimates for extremal values of the ratio R_k(G) = W(L^k(G))/W(G) where L^k(G) denotes the k-th iterated line graph of G. Hriňáková, Knor and Škrekovski [Art Discrete Appl. Math. 1 (2018) #P1.09] prove that for each k>2, the path P_n has the smallest value of the ratio R_k among all trees of large order n, and they conjecture that the same holds for the case k=2. We give a counterexample of every order n>21 to this conjecture.
Paper Structure (5 sections, 8 theorems, 29 equations, 3 figures)

This paper contains 5 sections, 8 theorems, 29 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and Notation
  3. Counterexamples homeomorphic to the star $K_{1,3}$
  4. Further homeomorphic classes
  5. Concluding remarks

Key Result

Theorem 1

tepeh2015 Among all connected graphs $G$ of order $n$, the ratio $R_1(G)$ is minimum for the star $S_n=K_{1,n-1}$.

Figures (3)

  • Figure 1: The tree $T_{3,4,5}$ and its second-order iterated line graph $L^2(T_{3,4,5})$.
  • Figure 2: The tree $U_{5}$. Deleting the hollow nodes leaves $Q_5$.
  • Figure 3: The second-order iterated line graph $L^2(U_{5})$. Deleting the hollow nodes leaves $L^2(Q_5)$.

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Conjecture 1
  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Lemma 3
  • Lemma 4
  • ...and 3 more