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On the multiplicatively weighted Harary index of composite graphs

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https://doi.org/10.3969/j.issn.0253-2778.2020.03.002
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  • Author Bio:

    TIAN Jing, female, born in 1993, Master candidate. Research field: Graph theory and its applications. E-mail: JingTian526@126.com

  • Corresponding author: PAN Xiangfeng
  • Received Date: 12 October 2018
  • Accepted Date: 30 October 2019
  • Rev Recd Date: 30 October 2019
  • Publish Date: 31 March 2020
    Abstract

  • Abstract

    Let HM(G) be the multiplicatively weighted Harary index of the molecular graph G, which is defined as HM(G)=∑{u,v}V(G)dG(u)dG(v)dG(u,v), where dG(u) is the degree of a vertex u∈V(G), and the dG(u,v) denotes the distance between u and v in G. We introduce four graph operations and obtain explicit formulas for the values of multiplicatively weighted Harary index of composite graphs generated by the four graph operations. Based on this, a lower and an upper bound is determined for the multiplicatively weighted Harary index among graphs in each of the four classes of composite graphs.

    Abstract

    Let HM(G) be the multiplicatively weighted Harary index of the molecular graph G, which is defined as HM(G)=∑{u,v}V(G)dG(u)dG(v)dG(u,v), where dG(u) is the degree of a vertex u∈V(G), and the dG(u,v) denotes the distance between u and v in G. We introduce four graph operations and obtain explicit formulas for the values of multiplicatively weighted Harary index of composite graphs generated by the four graph operations. Based on this, a lower and an upper bound is determined for the multiplicatively weighted Harary index among graphs in each of the four classes of composite graphs.
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    • References(26)
  • [1]
    BONDY J A, MURTY U S R. Graph Theory with Applications[M]. New York: Macmillan Press, 1976.
    [2]
    Buckley F, Harary F. Distance in Graphs[M]. Reading: Addison-Wesley, 1989.
    [3]
    GENG X Y, LI S C, ZHANG M. Extremal values on the eccentric distance sum of trees[J]. Discrete Appl. Math., 2013, 161: 2427-2439.
    [4]
    WANG H. The distances between internal vertices and leaves of a tree[J]. European J. Comb., 2014, 41: 79-99.
    [5]
    YEH Y,GUTMAN I. On the sum of all distances in composite graphs[J]. Discrete Math., 1994, 135: 359-365.
    [6]
    WIENER H. Structural determination of paran boiling point[J]. J. Amer. Chem. Soc., 1947, 69: 17-20.
    [7]
    GUTMAN I. A property of the wiener number and its modications[J]. Indian J. Chem., 1997, 36A: 128-132.
    [8]
    DOBRYNIN A, ENTRINGER R, GUTMAN I. Wiener index of trees: theory and applications[J]. Acta Appl. Math., 2001, 66: 211-249.
    [9]
    GUTMAN I, RADA J, ARAUJO O. The Wiener index of starlike trees and a related partial order[J]. MATCH Commun. Math. Comput. Chem., 2000, 42: 145-154.
    [10]
    NIKOLIC＇ S, KOVAC＇EVIC＇ G, MILICˇEVIC＇, et al. The Zagreb indices 30 years after[J]. Croat. Chem. Acta, 2003, 76: 113-124.
    [11]
    DOLIC＇ T. Vertex-weighted Wiener polynomials for composite graphs[J]. Ars Math. Contemp., 2008, 1: 66-80.
    [12]
    PLAVIC＇ D, NIKOLIC＇ S, TRINAJSTIC＇ N, et al. On the Harary index for the Characterization of chemical graphs[J]. J. Math. Chem., 1993, 12: 235-250.
    [13]
    IVANCIUC O, BALABAN S, BALABAN T. Reciprocal distance matrix related local vertex invariants and topological indices[J].J. Math. Chem., 1993, 12: 309-318.
    [14]
    DEVILLERS J, BALABAN A T. TopologicalIndices and Related Descriptors in QSAR and QSPR[M].Amsterdam: Gordon, Breach, 1999.
    [15]
    TODESCHINI R, CONSONNI V. Handbook of Molecular Descriptors[M]. Weinheim: Wiley-VCH, 2000.
    [16]
    ZHOU B, CAI X, TRINAJSTIC＇ N. On Harary index[J]. J. Math. Chem., 2008, 44: 611-618.
    [17]
    XU K, DAS K C. Extremal unicyclic and bicyclic graphs with respect to Harary index[J]. Bull. Malays. Math. Sci. Soc., 2013, 36: 373-383.
    [18]
    FENG L, LAN Y, LIU W,et al. Minimal Harary index of graphs with small parameters[J]. MATCH Commun. Math. Comput. Chem., 2016, 76: 23-42.
    [19]
    LI X, FAN Y. The connectivity and the Harary index of a graph[J]. Discrete Appl. Math., 2015, 181: 167-173.
    [20]
    BRUCKLER F M,DOLIC＇ T, GRAOVAC A, et al. On a class of distance-based molecular structure descriptors[J]. Chem. Phys. Lett., 2011, 503: 336-338.
    [21]
    ALIZADEH Y, IRANMANESH A,DOLIC＇ T. Additively weighted Harary index of some composite graphs[J]. Discrete Math., 2013, 313: 26-34.
    [22]
    DENG H, KRISHNAKUMARI B,VENKATAKRISHNAN B,et al. Multiplicatively weighted Harary index of graphs[J]. J. Comb. Optim., 2015, 30: 1125-1137.
    [23]
    LI S, ZHANG H. Some extremal properties of the multiplicatively weighted Harary index of a graph[J]. J. Comb. Optim., 2014, 31: 1-18.
    [24]
    LIU S QU J, LIN Y. Maximum additively weighted Harary index of graphs with given connectivity or matching number[J]. Int. J. Appl. Math. Stat., 2014, 52: 153-157.
    [25]
    KHOSRAVI B, RAMEZANI E. On the additively weighted harary index of some composite graphs[J]. Mathematics, 2017, 5 (1): 16.
    [26]
    ASHRAFI A R, DOLIC＇ T, HAMZEH A. The Zagreb coindices of graph operations[J]. Discrete Appl. Math., 2010, 158: 1571-1578.)
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    [1]
    BONDY J A, MURTY U S R. Graph Theory with Applications[M]. New York: Macmillan Press, 1976.
    [2]
    Buckley F, Harary F. Distance in Graphs[M]. Reading: Addison-Wesley, 1989.
    [3]
    GENG X Y, LI S C, ZHANG M. Extremal values on the eccentric distance sum of trees[J]. Discrete Appl. Math., 2013, 161: 2427-2439.
    [4]
    WANG H. The distances between internal vertices and leaves of a tree[J]. European J. Comb., 2014, 41: 79-99.
    [5]
    YEH Y,GUTMAN I. On the sum of all distances in composite graphs[J]. Discrete Math., 1994, 135: 359-365.
    [6]
    WIENER H. Structural determination of paran boiling point[J]. J. Amer. Chem. Soc., 1947, 69: 17-20.
    [7]
    GUTMAN I. A property of the wiener number and its modications[J]. Indian J. Chem., 1997, 36A: 128-132.
    [8]
    DOBRYNIN A, ENTRINGER R, GUTMAN I. Wiener index of trees: theory and applications[J]. Acta Appl. Math., 2001, 66: 211-249.
    [9]
    GUTMAN I, RADA J, ARAUJO O. The Wiener index of starlike trees and a related partial order[J]. MATCH Commun. Math. Comput. Chem., 2000, 42: 145-154.
    [10]
    NIKOLIC＇ S, KOVAC＇EVIC＇ G, MILICˇEVIC＇, et al. The Zagreb indices 30 years after[J]. Croat. Chem. Acta, 2003, 76: 113-124.
    [11]
    DOLIC＇ T. Vertex-weighted Wiener polynomials for composite graphs[J]. Ars Math. Contemp., 2008, 1: 66-80.
    [12]
    PLAVIC＇ D, NIKOLIC＇ S, TRINAJSTIC＇ N, et al. On the Harary index for the Characterization of chemical graphs[J]. J. Math. Chem., 1993, 12: 235-250.
    [13]
    IVANCIUC O, BALABAN S, BALABAN T. Reciprocal distance matrix related local vertex invariants and topological indices[J].J. Math. Chem., 1993, 12: 309-318.
    [14]
    DEVILLERS J, BALABAN A T. TopologicalIndices and Related Descriptors in QSAR and QSPR[M].Amsterdam: Gordon, Breach, 1999.
    [15]
    TODESCHINI R, CONSONNI V. Handbook of Molecular Descriptors[M]. Weinheim: Wiley-VCH, 2000.
    [16]
    ZHOU B, CAI X, TRINAJSTIC＇ N. On Harary index[J]. J. Math. Chem., 2008, 44: 611-618.
    [17]
    XU K, DAS K C. Extremal unicyclic and bicyclic graphs with respect to Harary index[J]. Bull. Malays. Math. Sci. Soc., 2013, 36: 373-383.
    [18]
    FENG L, LAN Y, LIU W,et al. Minimal Harary index of graphs with small parameters[J]. MATCH Commun. Math. Comput. Chem., 2016, 76: 23-42.
    [19]
    LI X, FAN Y. The connectivity and the Harary index of a graph[J]. Discrete Appl. Math., 2015, 181: 167-173.
    [20]
    BRUCKLER F M,DOLIC＇ T, GRAOVAC A, et al. On a class of distance-based molecular structure descriptors[J]. Chem. Phys. Lett., 2011, 503: 336-338.
    [21]
    ALIZADEH Y, IRANMANESH A,DOLIC＇ T. Additively weighted Harary index of some composite graphs[J]. Discrete Math., 2013, 313: 26-34.
    [22]
    DENG H, KRISHNAKUMARI B,VENKATAKRISHNAN B,et al. Multiplicatively weighted Harary index of graphs[J]. J. Comb. Optim., 2015, 30: 1125-1137.
    [23]
    LI S, ZHANG H. Some extremal properties of the multiplicatively weighted Harary index of a graph[J]. J. Comb. Optim., 2014, 31: 1-18.
    [24]
    LIU S QU J, LIN Y. Maximum additively weighted Harary index of graphs with given connectivity or matching number[J]. Int. J. Appl. Math. Stat., 2014, 52: 153-157.
    [25]
    KHOSRAVI B, RAMEZANI E. On the additively weighted harary index of some composite graphs[J]. Mathematics, 2017, 5 (1): 16.
    [26]
    ASHRAFI A R, DOLIC＇ T, HAMZEH A. The Zagreb coindices of graph operations[J]. Discrete Appl. Math., 2010, 158: 1571-1578.)
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