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Sufficient conditions for Hamiltonianity in terms of the Zeroth-order General Randić Index

Shuai Wang, Lihong Cui

Abstract

For a (molecular) graph $G$ and any real number $α\ne 0$ , the zero-order general Randić index , denote by $^0R_α$, is defined by the following equation: \begin{align*} {^0R_α} (G) =\sum_{v\in G}d_G (v) ^α (α\in \mathbb{R}-\left\{0\right\}) . \end{align*} In this paper, we use this index to give sufficient conditions for a graph $G$ to satisfy the Hamiltonian (or $k$-Hamiltonian) property, and show that none of these conditions can be dropped. Finally we give similar results for the case when $G$ is a balanced bipartite graph.

Sufficient conditions for Hamiltonianity in terms of the Zeroth-order General Randić Index

Abstract

For a (molecular) graph and any real number , the zero-order general Randić index , denote by , is defined by the following equation: \begin{align*} {^0R_α} (G) =\sum_{v\in G}d_G (v) ^α (α\in \mathbb{R}-\left\{0\right\}) . \end{align*} In this paper, we use this index to give sufficient conditions for a graph to satisfy the Hamiltonian (or -Hamiltonian) property, and show that none of these conditions can be dropped. Finally we give similar results for the case when is a balanced bipartite graph.

Paper Structure

This paper contains 4 sections, 15 theorems, 67 equations.

Table of Contents

  1. Introduction and Notation
  2. Proof of Theorem \ref{['the:th2.3']}
  3. Proof of Theorem \ref{['the:th2.6']} to \ref{['the:th2.8']}
  4. Proof of Theorem \ref{['the:th2.12']}

Key Result

Theorem 1.1

(An et al. An-2018) Let $G$ be a simple connected graph of order $n\ge 5$, $0\le k\le n-3$ and If $M_1 (G) >h_1 (n,k)$, then $G$ is $k$-Hamiltonian. Moreover, ${M_1} (G) =h (n,k)$ if and only if $G\cong K_{k+1}\vee (K_1\cup K_{n-k-2})$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Lemma 2.1
  • ...and 18 more