On the extremal values of the eccentric distance sum of trees with a given domination number

Abstract Let G be a simple connected graph. The eccentric distance sum (EDS) of G is defined as ξ d ( G ) = ∑ v ∈ V e G ( v ) D G ( v ) , where e G ( v ) is the eccentricity of the vertex v and D G ( v ) = ∑ u ∈ V d G ( u , v ) is the sum of all distances from the vertex v . In this paper, the trees having the maximal EDS among n -vertex trees with maximum degree Δ and among those with domination number 3 are characterized. The trees having the maximal or minimal EDS among n -vertex trees with independence number α and the trees having the maximal EDS among n -vertex trees with matching number m are also determined.

[1]  Guihai Yu,et al.  On the eccentric distance sum of trees and unicyclic graphs , 2011 .

[2]  P. Dankelmann,et al.  The Average Eccentricity of a Graph and its Subgraphs , 2022 .

[3]  Guihai Yu,et al.  On the eccentric distance sum of graphs , 2011 .

[4]  A. K. Madan,et al.  Eccentric Connectivity Index: A Novel Highly Discriminating Topological Descriptor for Structure-Property and Structure-Activity Studies , 1997 .

[5]  Teresa W. Haynes,et al.  Extremal graphs for inequalities involving domination parameters , 2000, Discret. Math..

[6]  I. Gutman,et al.  Eccentric Connectivity Index of Chemical Trees , 2011, 1104.3206.

[7]  Michael S. Jacobson,et al.  On graphs having domination number half their order , 1985 .

[8]  O. Ore.  Theory of Graphs , 1962 .

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

[10]  Ali Reza Ashrafi,et al.  The eccentric connectivity index of nanotubes and nanotori , 2011, J. Comput. Appl. Math..

[11]  Roger C. Entringer,et al.  Distance in graphs , 1976 .

[12]  A. K. Madan,et al.  Application of Graph Theory: Relationship of Eccentric Connectivity Index and Wiener's Index with Anti-inflammatory Activity , 2002 .

[13]  Shuchao Li,et al.  Extremal values on the eccentric distance sum of trees , 2012, Discret. Appl. Math..

[14]  F. Harary,et al.  On the corona of two graphs , 1970 .

[15]  Simon Mukwembi,et al.  On the eccentric connectivity index of a graph , 2011, Discret. Math..

[16]  Harish Dureja,et al.  Predicting anti-HIV-1 activity of 6-arylbenzonitriles: computational approach using superaugmented eccentric connectivity topochemical indices. , 2008, Journal of molecular graphics & modelling.

[17]  Wolfgang Linert,et al.  Trees with Extremal Hyper-Wiener Index: Mathematical Basis and Chemical Applications , 1997, J. Chem. Inf. Comput. Sci..

[18]  Kexiang Xu,et al.  A short and unified proof of Yu et al.ʼs two results on the eccentric distance sum , 2011 .

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

[20]  Shenggui Zhang,et al.  Further results on the eccentric distance sum , 2012, Discret. Appl. Math..

[21]  Sunil Gupta,et al.  Eccentric distance sum: A novel graph invariant for predicting biological and physical properties , 2002 .

[22]  Aleksandar Ilic,et al.  Eccentric connectivity index , 2011, 1103.2515.

[23]  Kexiang Xu,et al.  Extremal energies of trees with a given domination number , 2011 .