Reciprocal degree distance and graph properties

Abstract The reciprocal degree distance (RDD), defined for a connected graph G as vertex-degree- weighted sum of the reciprocal distances, that is, R D D ( G ) = ∑ u ≠ v d G ( u ) + d G ( v ) d G ( u , v ) , where d G ( u ) is the degree of the vertex u in the graph G and d G ( u , v ) denotes the distance between two vertices u and v in the graph G . The reciprocal degree distance is a weight version of the Harary index, just as the degree distance is a weight version of the Wiener index. Finding sufficient conditions for graphs possessing certain properties is an important and meaningful problem. In this paper, we give sufficient conditions for a graph to be k -connected or β -deficient in terms of the reciprocal degree distance.

[1]  Ivan Gutman,et al.  A PROPERTY OF THE WIENER NUMBER AND ITS MODIFICATIONS , 1997 .

[2]  Xiaomin Zhu,et al.  Wiener index, Harary index and graph properties , 2017, Discret. Appl. Math..

[3]  Ivan Gutman,et al.  Selected properties of the Schultz molecular topological index , 1994, J. Chem. Inf. Comput. Sci..

[4]  Yaser Alizadeh,et al.  Additively weighted Harary index of some composite graphs , 2013, Discret. Math..

[5]  Kinkar Chandra Das,et al.  Weighted Harary indices of apex trees and k-apex trees , 2015, Discret. Appl. Math..

[6]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[7]  Andrey A. Dobrynin,et al.  Degree Distance of a Graph: A Degree Analog of the Wiener Index , 1994, J. Chem. Inf. Comput. Sci..

[8]  Ioan Tomescu.  Ordering connected graphs having small degree distances , 2010, Discret. Appl. Math..

[9]  I. Gutman,et al.  Mathematical Concepts in Organic Chemistry , 1986 .

[10]  I. Gutman,et al.  Wiener Index of Trees: Theory and Applications , 2001 .

[11]  Shenggui Zhang,et al.  On the reciprocal degree distance of graphs , 2012, Discret. Appl. Math..

[12]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[13]  Ali Reza Ashrafi,et al.  Calculating the edge Wiener and edge Szeged indices of graphs , 2011, J. Comput. Appl. Math..

[14]  P. Alam.  ‘O’ , 2021, Composites Engineering: An A–Z Guide.

[15]  H. Wiener.  Structural determination of paraffin boiling points. , 1947, Journal of the American Chemical Society.

[16]  Douglas R. Woodall,et al.  Best Monotone Degree Conditions for Graph Properties: A Survey , 2015, Graphs Comb..

[17]  Ruifang Liu,et al.  Wiener index on traceable and hamiltonian graphs , 2016 .

[18]  N. Trinajstic,et al.  On the Harary index for the characterization of chemical graphs , 1993 .

[19]  Lihui Yang.  WIENER INDEX AND TRACEABLE GRAPHS , 2012, Bulletin of the Australian Mathematical Society.

[20]  B. Bollobás,et al.  Extremal Graph Theory , 2013 .

[21]  Peter Dankelmann,et al.  Degree distance of unicyclic and bicyclic graphs , 2011, Discret. Appl. Math..

[22]  M. Vijayaragavan,et al.  Reciprocal degree distance of product graphs , 2014, Discret. Appl. Math..

[23]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[24]  Shuchao Li,et al.  On the reformulated reciprocal sum-degree distance of graph transformations , 2015, Discret. Appl. Math..

[25]  Elwood S. Buffa,et al.  Graph Theory with Applications , 1977 .

[26]  Hongbo Hua,et al.  On Harary Index and Traceable Graphs , 2013 .