Wiener Index and Traceable Graphs

Abstract

In this short paper, we show that, with three exceptions, if the Wiener index of a connected graph of order n is at most (n + 5)(n − 2)/2, then it is traceable.2010 Mathematics subject classification: primary 05C12; secondary 05C07, 05C45.

Some sufficient conditions for Hamiltonian property in terms of Wiener-type invariants

“…We can get the following sufficient conditions in terms of the Wiener index, Harary index, hyper-Wiener index, modified Wiener index, respectively, for a connected graph to be traceable. Note that the following Corollaries 9 and 10 are also given in [17] and [10], respectively, but there is a flaw in the proof of their theorems for 4 ≤ n ≤ 8.…”

“…Concerning the existence of Hamiltonian path or Hamiltonian cycle, there are many famous sufficient conditions in terms of its vertex degrees, such as Dirac's condition [3], Chvátal's condition [2] and so on. Recently, Yang [17] and Hua and Wang [10] gave a sufficient condition for a connected graph to be traceable by using its Wiener index and Harary index, respectively. Zeng [18] also gave a sufficient condition, in terms of Harary index, for a connected bipartite graph to be Hamiltonian.…”

Wiener Index on Traceable and Hamiltonian graphs

“…In Sections 2-3, we give sufficient conditions for a graph to be traceable and Hamiltonian in terms of the Wiener index and the complement of the graph, which correct and extend the result of Yang [10]. In Section 4, we present sufficient conditions for a bipartite graph to be traceable and Hamiltonian in terms of its Wiener index and quasicomplement.…”

“…Many necessary or sufficient conditions have been given for a graph to be traceable or Hamiltonian. Recently, some sufficient spectral conditions involving the Wiener index and distance spectral radius for a graph to be Hamiltonian and traceable have been given in [4][5][6]10].…”

Hosoya and Harary Polynomials ofTOX(n),RTOX(n),TSLChen1, Mehboob2, Ahmad3 et al. 2019Discrete Dynamics in Nature and Society

“…It plays an important role in the so-called inverse structureproperty relationship problems [13]. For more details about this topological polynomial and index, please see the paper series and the references therein [14][15][16][17][18][19][20][21][22]. Note that the first derivative of the Hosoya polynomial at = 1 is equal to the Wiener index:…”