Wiener index and Harary index on Hamilton-connected graphs with large minimum degree

2018 ◽  
Vol 247 ◽  
pp. 180-185 ◽  
Author(s):  
Qiannan Zhou ◽  
Ligong Wang ◽  
Yong Lu
Keyword(s):  
Minimum Degree ◽  
Wiener Index ◽  
Harary Index ◽  
Connected Graphs ◽  
Hamilton Connected

scholarly journals Sufficient Conditions for Hamiltonicity of Graphs with Respect to Wiener Index, Hyper-Wiener Index, and Harary Index

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Guisheng Jiang ◽  
Lifang Ren ◽  
Guidong Yu
Keyword(s):  
Minimum Degree ◽  
Sufficient Conditions ◽  
Wiener Index ◽  
Bipartite Graphs ◽  
Harary Index ◽  
Connected Graphs ◽  
Hamilton Connected

In this paper, with respect to the Wiener index, hyper-Wiener index, and Harary index, it gives some sufficient conditions for some graphs to be traceable, Hamiltonian, Hamilton-connected, or traceable for every vertex. Firstly, we discuss balanced bipartite graphs with δG≥t, where δG is the minimum degree of G, and gain some sufficient conditions for the graphs to be traceable or Hamiltonian, respectively. Secondly, we discuss nearly balanced bipartite graphs with δG≥t and present some sufficient conditions for the graphs to be traceable. Thirdly, we discuss graphs with δG≥t and obtain some conditions for the graphs to be traceable or Hamiltonian, respectively. Finally, we discuss t-connected graphs and provide some conditions for the graphs to be Hamilton-connected or traceable for every vertex, respectively.

A constructive characterization of contraction critical 8-connected graphs with minimum degree 9

2019 ◽  
Vol 342 (11) ◽  
pp. 3047-3056
Author(s):  
Chengfu Qin ◽  
Weihua He ◽  
Kiyoshi Ando
Keyword(s):  
Minimum Degree ◽  
Connected Graphs

scholarly journals Wiener–Hosoya Matrix of Connected Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 359
Author(s):  
Hassan Ibrahim ◽  
Reza Sharafdini ◽  
Tamás Réti ◽  
Abolape Akwu
Keyword(s):  
Lower Bounds ◽  
Molecular Graph ◽  
Wiener Index ◽  
Vertex Degree ◽  
Upper And Lower Bounds ◽  
Connected Graphs ◽  
Matrix Description ◽  
Vertex Set

Let G be a connected (molecular) graph with the vertex set V(G)={v1,⋯,vn}, and let di and σi denote, respectively, the vertex degree and the transmission of vi, for 1≤i≤n. In this paper, we aim to provide a new matrix description of the celebrated Wiener index. In fact, we introduce the Wiener–Hosoya matrix of G, which is defined as the n×n matrix whose (i,j)-entry is equal to σi2di+σj2dj if vi and vj are adjacent and 0 otherwise. Some properties, including upper and lower bounds for the eigenvalues of the Wiener–Hosoya matrix are obtained and the extremal cases are described. Further, we introduce the energy of this matrix.

Spectral analogues of Erdős’ theorem on Hamilton-connected graphs

2019 ◽  
Vol 340 ◽  
pp. 242-250
Author(s):  
Jia Wei ◽  
Zhifu You ◽  
Hong-Jian Lai
Keyword(s):  
Connected Graphs ◽  
Hamilton Connected

scholarly journals Hosoya and Harary Polynomials of TOX(n),RTOX(n),TSL(n) and RTSL(n)

2019 ◽  
Vol 2019 ◽  
pp. 1-18 ◽  
Author(s):  
Lian Chen ◽  
Abid Mehboob ◽  
Haseeb Ahmad ◽  
Waqas Nazeer ◽  
Muhammad Hussain ◽  
...  
Keyword(s):  
Topological Index ◽  
Molecular Descriptor ◽  
Wiener Index ◽  
Vital Role ◽  
Harary Index ◽  
Chemical Graph ◽  
Chemical Graph Theory ◽  
Hosoya Polynomial ◽  
Molecular Branching ◽  
Path Number

In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. In 1947, Wiener introduced “path number” which is now known as Wiener index and is the oldest topological index related to molecular branching. Hosoya polynomial plays a vital role in determining Wiener index. In this report, we computed the Hosoya and the Harary polynomials for TOX(n),RTOX(n),TSL(n), and RTSL(n) networks. Moreover, we computed serval distance based topological indices, for example, Wiener index, Harary index, and multiplicative version of wiener index.

On the maximum connective eccentricity index among k-connected graphs

2020 ◽  
pp. 2150002
Author(s):  
Fazal Hayat
Keyword(s):  
Minimum Degree ◽  
Independence Number ◽  
Connected Graphs ◽  
Eccentricity Index

The connective eccentricity index (CEI for short) of a graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the degree of [Formula: see text] and [Formula: see text] is the eccentricity of [Formula: see text] in [Formula: see text]. In this paper, we characterize the unique graphs with maximum CEI from three classes of graphs: the [Formula: see text]-vertex graphs with fixed connectivity and diameter, the [Formula: see text]-vertex graphs with fixed connectivity and independence number, and the [Formula: see text]-vertex graphs with fixed connectivity and minimum degree.

On distance Laplacian spectrum (energy) of graphs

2020 ◽  
Vol 12 (05) ◽  
pp. 2050061 ◽  
Author(s):  
Hilal A. Ganie
Keyword(s):  
Chromatic Number ◽  
Wiener Index ◽  
Laplacian Spectrum ◽  
Connected Graphs ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
Number Formula ◽  
Energy Formula ◽  
Average Transmission ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] of order [Formula: see text] having distance Laplacian eigenvalues [Formula: see text], the distance Laplacian energy [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the Wiener index of [Formula: see text]. We obtain the distance Laplacian spectrum of the joined union of graphs [Formula: see text] in terms of their distance Laplacian spectrum and the spectrum of an auxiliary matrix. As application, we obtain the distance Laplacian spectrum of the lexicographic product of graphs. We study the distance Laplacian energy of connected graphs with given chromatic number [Formula: see text]. We show that among all connected graphs with chromatic number [Formula: see text] the complete [Formula: see text]-partite graph has the minimum distance Laplacian energy. Further, we discuss the distribution of distance Laplacian eigenvalues around average transmission degree [Formula: see text].

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