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Journal of Combinatorial Optimization

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21-10-2019

On the extremal graphs with respect to the total reciprocal edge-eccentricity

Authors: Lifang Zhao, Hongshuai Li, Yuping Gao

Published in: Journal of Combinatorial Optimization | Issue 1/2020

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Abstract

The total reciprocal edge-eccentricity of a graph G is defined as \(\xi ^{ee}(G)=\sum _{u\in V_G}\frac{d_G(u)}{\varepsilon _G(u)}\), where \(d_G(u)\) is the degree of u and \(\varepsilon _G(u)\) is the eccentricity of u. In this paper, we first characterize the unique graph with the maximum total reciprocal edge-eccentricity among all graphs with a given number of cut vertices. Then we determine the k-connected bipartite graphs of order n with diameter d having the maximum total reciprocal edge-eccentricity. Finally, we identify the unique tree with the minimum total reciprocal edge-eccentricity among the n-vertex trees with given degree sequence.

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Behmaram A, Yousefi-Azari H, Ashrafi AR (2012) Wiener polarity index of fullerenes and hexagonal systems. Appl Math Lett 25(10):1510–1513MathSciNetMATH

Bondy JA, Murty USR (2008) Graph theory. In: Axler S, Ribet KA (eds) Graduate texts in mathematics, vol 244. Springer, New York

Dankelmann P, Goddard W, Swart CS (2004) The average eccentricity of a graph and its subgraphs. Util Math 65:41–51MathSciNetMATH

Dobrynin A, Kochetova AA (1994) Degree distance of a graph: a degree analogue of the Wiener index. J Chem Inf Comput Sci 34:1082–1086

Dobrynin AA, Entriger R, Gutman I (2001) Wiener index of trees: theory and applications. Acta Appl Math 66:211–249MathSciNetMATH

Dobrynin AA, Gutman I, Klavžar S, Žigert P (2002) Wiener index of hexagonal systems. Acta Appl Math 72:247–294MathSciNetMATH

Entringer RC, Jackson DE, Snyder DA (1976) Distance in graphs. Czech Math J 26:283–296MathSciNetMATH

Geng XY, Li SC, Zhang M (2013) Extremal values on the eccentric distance sum of trees. Discrete Appl Math 161:2427–2439MathSciNetMATH

Georgakopoulos A, Wagner S (2017) Hitting times, cover cost, and the Wiener index of a tree. J Graph Theory 84(3):311–326MathSciNetMATH

Gupta S, Singh M, Madan AK (2000) Connective eccentricity index: a novel topological descriptor for predicting biological activity. J Mol Graph Model 18:18–25

Gupta S, Singh M, Madan AK (2002a) Application of graph theory: relationship of eccentric connectivity index and Wiener’s index with anti-inflammatory activity. J Math Anal Appl 266:259–268MathSciNetMATH

Gupta S, Singh M, Madan AK (2002b) Eccentric distance sum: a novel graph invariant for predicting biological and physical properties. J Math Anal Appl 275(1):386–401MathSciNetMATH

Gutman I (1994) Selected properties of the Schultz molecular topological index. J Chem Inf Comput Sci 34:1087–1089

Gutman I, Wagner S (2012) The matching energy of a graph. Discrete Appl Math 160(15):2177–2187MathSciNetMATH

Hosoya H (1971) Topological index. Newly proposed quautity characterizing the topological nature of structure of isomers of saturated hydrocarbons. Bull Chem Soc Jpn 44:2332–2337

Hosoya H (1972) Topological index as strong sorting device for coding chemical structure. J Chem Doc 12:181–183

Hou AL, Li SC, Song LZ, Wei B (2011) Sharp bounds for Zagreb indices of maximal outerplanar graphs. J Comb Optim 22(2):252–269MathSciNetMATH

Huang J, Li SC, Li XC (2016a) The normalized Laplacian, degree-Kirchhoff index and spanning trees of the linear polyomino chains. Appl Math Comput 289:324–334MathSciNetMATH

Huang J, Li SC, Sun LQ (2016b) The normalized Laplacians, degree-Kirchhoff index and the spanning trees of linear hexagonal chains. Discrete Appl Math 207:67–79MathSciNetMATH

Ilić A (2012) On the extremal properties of the average eccentricity. Comput Math Appl 64:2877–2885MathSciNetMATH

Knor M, Škrekovski R, Tepeh A (2016) Mathematical aspects of Wiener index. Ars Math Contemp 11:327–352MathSciNetMATH

Kumar V, Sardana S, Madan AK (2004) Predicting anti-HIV activity of 2,3-diaryl-1,3 thiazolidin-4-ones: computational approach using reformed eccentric connectivity index. J Mol Model 10:399–407

Li SC (2017) Sharp bounds on the eccentric distance sum of graphs. In: Gutman I, Furtula B, Das KC, Milovanović E, Milovanović I (eds) Bounds in chemical graph theory-mainstreams. University Kragujevac, Kragujevac, pp 207–237

Li XL, Shi YT (2008) A survey on the Randić index. MATCH Commun Math Comput Chem 59:127–156MathSciNetMATH

Li SC, Song YB (2014) On the sum of all distances in bipartite graphs. Discrete Appl Math 169:176–185MathSciNetMATH

Li SC, Wu YY (2016a) On the extreme eccentric distance sum of graphs with some given parameters. Discrete Appl Math 206:90–99MathSciNetMATH

Li SC, Zhao LF (2016b) On the extremal total reciprocal edge-eccentricity of trees. J Math Anal Appl 433:587–602MathSciNetMATH

Li SC, Zhang HH (2017) Proofs of three conjectures on the quotients of the (revised) Szeged index and the Wiener index and beyond. Discrete Math 340(3):311–324MathSciNetMATH

Li SC, Zhang M, Yu GH, Feng LH (2012) On the extremal values of the eccentric distance sum of trees. J Math Anal Appl 390:99–112MathSciNetMATH

Li SC, Wu YY, Sun LL (2015) On the minimum eccentric distance sum of bipartite graphs with some given parameters. J Math Anal Appl 430:1149–1162MathSciNetMATH

Li HS, Li SC, Zhang HH (2017) On the maximal connective eccentricity index of bipartite graphs with some given parameters. J Math Anal Appl 454:453–467MathSciNetMATH

Liu RF, Du X, Jia HC (2016) Wiener index on traceable and Hamiltonian graphs. Bull Aust Math Soc 94:362–372MathSciNetMATH

Mohar B (2015) Median eigenvalues and the HOMO-LUMO index of graphs. J Combin Theory Ser B 112:78–92MathSciNetMATH

Nikiforov V, Agudelo N (2017) On the minimum trace norm/energy of \((0,1)\)-matrices. Linear Algebra Appl 526:42–59MathSciNetMATH

Sedlar J (2012) On augmented eccentric connectivity index of graphs and trees. MATCH Commun Math Comput Chem 68(1):325–342MathSciNetMATH

Sharma V, Goswami R, Madan AK (1997) Eccentric connectivity index: a novel highly discriminating topological descriptor for structure property and structure activity studies. J Chem Inf Comput Sci 37:273–282

Tang L, Wang X, Liu WJ, Feng LH (2017) The extremal values of connective eccentricity index for trees and unicyclic graphs. Int J Comput Math 94(3):437–453MathSciNetMATH

Wang X, Tang L, Chen XS, Li MS, Li Y (2018) On the connective eccentricity index of graphs with fixed clique number. Ars Combin 138:105–117MathSciNetMATH

Wiener H (1947a) Correlation of heat of isomerization and difference in heat of vaporization of isomers among paraffin hydrocaibons. J Am Chem Soc 69:2636–2638

Wiener H (1947b) Influence of interatomic forces on paraffin properties. J Chem Phys 15:766–767

Wiener H (1947c) Structural determination of paraffin boiling points. J Am Chem Soc 69:17–20

Wiener H (1948) Vapour–pressure–temperature relations among the branched paraffin hydrocarbons. J Chem Phys 15:425–430

Xu KX (2017) Some bounds on the eccentricity-based topological indices of graphs. In: Gutman I, Furtula B, Das KC, Milovanović E, Milovanović I (eds) Bounds in chemical graph theory-mainstreams. University Kragujevac, Kragujevac, pp 189–205

Xu KX, Das KCh, Liu HQ (2016) Some extremal results on the connective eccentricity index of graphs. J Math Anal Appl 433(2):803–817MathSciNetMATH

Yu GH, Feng LH (2013) On connective eccentricity index of graphs. MATCH Commun Math Comput Chem 69:611–628MathSciNetMATH

Yu GH, Qu H, Tang L, Feng LH (2014) On connective eccentricity index of trees and unicyclic graphs with given diameter. J Math Anal Appl 420(2):1776–1786MathSciNetMATH

Zhang HH, Chen J, Li SC (2017) On the quotients between the (revised) Szeged index and Wiener index of graphs. Discrete Math Theor Comput Sci 19, no. 1, Paper No. 12

Title: On the extremal graphs with respect to the total reciprocal edge-eccentricity
Authors: Lifang Zhao
Hongshuai Li
Yuping Gao
Publication date: 21-10-2019
Publisher: Springer US
Published in: Journal of Combinatorial Optimization / Issue 1/2020
Print ISSN: 1382-6905
Electronic ISSN: 1573-2886
DOI: https://doi.org/10.1007/s10878-019-00458-2

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