On the Transmission-Based Graph Topological Indices
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Authors: R. SHARAFDINI AND T. RéTI
DOI: 10.46793/KgJMat2001.041S
Abstract:
The distance d(u,v) between vertices u and v of a connected graph G is equal to the number of edges in a minimal path connecting them. The transmission of a vertex v is defined by σ(v) = ∑ u∈V (G)d(v,u). A topological index is said to be a transmission-based topological index (TT index) if it includes the transmissions σ(u) of vertices of G. Because σ(u) can be derived from the distance matrix of G, it follows that transmission-based topological indices form a subset of distance-based topological indices. So far, relatively limited attention has been paid to TT indices, and very little systematic studies have been done. In this paper our aim was i) to define various types of transmission-based topological indices ii) establish lower and upper bounds for them, and iii) determine a family of graphs for which these bounds are best possible. Additionally, it has been shown in examples that using a group theoretical approach the transmission-based topological indices can be easily computed for a particular set of regular, vertex-transitive, and edge-transitive graphs. Finally, it is demonstrated that there exist TT indices which can be successfully applied to predict various physicochemical properties of different organic compounds. Some of them give better results and have a better discriminatory power than the most popular degree-based and distance-based indices (Randić, Wiener, Balaban indices).
Keywords:
Graph distance, topological index, transmission.
References:
[1] M. O. Albertson, The irregularity of a graph, Ars Combin. 46 (1997), 219–225.
[2] M. Aouchiche and P. Hansen, Distance spectra of graphs: a survey, Linear Algebra Appl. 458 (2014), 301–386.
[3] M. Aouchiche and P. Hansen, On a conjecture about Szeged index, European J. Combin. 31 (2010), 1662–1666.
[4] A. T. Balaban, Highly discriminating distance based numerical descriptor, Chemical Physics Letters 89 (1982), 399–404.
[5] A. T. Balaban, Topological indices based on topological distances in molecular graphs, Pure and Applied Chemistry 55 (1983), 199–206.
[6] A. T. Balaban, P. V. Khadikar and S. Aziz, Comparison of topological indices based on iterated sum versus product operations, Iranian Journal of Mathematical Chemistry 1 (2010), 43–67.
[7] F. K. Bell, A note on the irregularity of graphs, Linear Algebra Appl. 161 (1992), 45–54.
[8] M. R. Darafsheh, Computation of topological indices of some graphs, Acta Appl. Math. 110 (2010), 1225–1235.
[9] K. C. Das and I. Gutman, Estimating the Wiener index by means of number of vertices, number of edges, and diameter, MATCH Commun. Math. Comput. Chem. 64 (2010), 647–660.
[10] H. Deng, On the sum-Balaban index, MATCH Commun. Math. Comput. Chem. 66 (2011), 273–284.
[11] A. Dobrynin and A. A. Kochetova, Degree distance of a graph: a degree analogue of the Wiener index, Journal of Chemical Information and Computer Sciences 34 (1994), 1082–1086.
[12] A. Dobrynin and I. Gutman, On a graph invariant related to the sum of all distances in a graph, Publ. Inst. Math. (Beograd) (N.S.) 56 (1994), 18–22.
[13] A. Dobrynin, R. Entringer and I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math. 66 (2001), 200–249.
[14] J. K. Doyle and J. E. Graver, Mean distance in a graph, Discrete Math. 17 (1977), 147–154.
[15] J. D. Dixon and B. Mortimer, Permutation Groups, Springer-Verlag, New York, 1996.
[16] R. C. Entringer, D. E. Jackson and D. A. Snyder, Distance in graphs, Czechoslovak Math. J. 26 (1976), 283–296.
[17] G. H. Fath-Tabar, B. Furtula and I. Gutman, A new geometric-arithmetic index, J. Math. Chem. 47 (2014), 477–468.
[18] M. Ghorbani, Remarks on the Balaban index, Serdica J. Comput. 7 (2013), 25–34.
[19] C. Godsil, G. Royle, Algebraic Graph Theory, Springer-Verlag New York, 2001.
[20] S. Gupta, M. Singh and A. K. Madan, Eccentric distance sum: a novel graph invariant for predicting biological and physical properties, J. Math. Anal. Appl. 275 (2002), 386–401.
[21] I. Gutman and Y-N. Yeh, The sum of all distances in bipartite graphs, Math. Slovaca 45 (1995), 327–334.
[22] I. Gutman and K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004), 83–92.
[23] I. Gutman, Degree–based topological indices, Croatica Chemica Acta 86 (2013), 351–361.
[24] I. Gutman, Selected properties of the Schultz molecular topological index, Journal of Chemical Information and Computer Sciences 34 (1994), 1087–1089.
[25] A. Ilić and D. Stevanović, On comparing Zagreb indices, MATCH Commun. Math. Comput. Chem. 62 (2009), 681–687.
[26] A. Ilić, S. Klavžar and M. Milanović, On distance-balanced graphs, European J. Combin. 31 (2010), 733–737.
[27] A. Ilić, G. Yub and L. Fengc, On the eccentric distance sum of graphs, J. Math. Anal. Appl. 381 (2011), 590–600.
[28] M. H. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi and S. G. Wagner, Some new results on degree-based graph invariants, European J. Combin. 30 (2009), 1149–1163.
[29] S. Klavžar, A. Rajapakse and I. Gutman, The Szeged and the Wiener index of graphs, Appl. Math. Lett. 9 (1996), 45–49.
[30] H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl and R. E. Smalley, C60: Buckminsterfullerene, Nature 318 (1985), 162–163.
[31] B. Liu and Z. You, A survey on comparing Zagreb indices, MATCH Commun. Math. Comput. Chem. 65 (2011), 581–593.
[32] H. Liu and X. F. Pan, On the Wiener index of trees with fixed diameter, MATCH Commun. Math. Comput. Chem. 60 (2008), 85–94.
[33] A. Miličević and S. Nikolić, On variable Zagreb indices, Croatica Chemica Acta 77 (2004), 97–101.
[34] R. Mohammadyari and M. R. Darafsheh, Topological indices of the Kneser graph KGn;k, Filomat 26 (2012), 665–672.
[35] M. J. Nadjafi-Arani, H. Khodashenas and A. R. Ashrafi, On the differences between Szeged and Wiener indices of graphs, Discrete Math. 311 (2011), 2233–2237.
[36] S. Nikolić, G. Kovačević, M. Miličević and N. Trinajstić, The Zagreb indices 30 years after, Croatica Chemica Acta 76 (2003), 113–124.
[37] V. Nikiforov, Eigenvalues and degree deviation in graphs, Linear Algebra Appl. 414 (2006), 347–360.
[38] T. Pisanski and M. Randić, Use of Szeged index and the revised Szeged index for measuring the network bipartivity, Discrete Appl. Math. 158 (2010), 1936–1944.
[39] H. S. Ramane and A. S. Yalnaik, Status connectivity indices of graphs and its applications to the boiling point of benzenoid hydrocarbons, J. Appl. Math. Comput. 1-2 (2017), DOI: 10.1007/s12190-016-1052-5.
[40] M. Randić, On characterization of molecular branching, Journal of the American Chemical Society 97 (1975), 6609–6615.
[41] B. Ren, A new topological index for QSPR of alcanes, Journal of Chemical Information and Computer Sciences 39 (1999), 139–143.
[42] M. Shamsipur, B. Hemmateenejad and M. Akhound, Highly correlating distance connectivity-based topoligical indices. 1: QSPR studies of alkanes, Bull. Korean Chem. Soc. 25 (2004), 253–259.
[43] A. I. Tomescu, Unicyclic and bicyclic graphs having minimum degree distance, Discrete Appl. Math. 156 (2008), 125–130.
[44] D. B. West, Introduction to Graph Theory, Prentic-Hall, Upper Saddle River, New Jersey, 1996.
[45] D. Vukičević and B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem. 46 (2009), 1369–1376.
[46] B. A. Xavier, E. Suresh and I. Gutman, Counting relations for general Zagreb indices, Kragujevac J. Math. 38 (2014), 95–103.
[47] R. Xing and B. Zhou, On the revised Szeged index, Discrete Appl. Math. 159 (2011), 69–78.
[48] B. Zhou and N. Trinajstić, On general sum-connectivity index, J. Math. Chem. 47 (2010), 2010–2018.
[49] L. Zhong, The harmonic index for graphs, Appl. Math. Lett. 25 (2012), 561–566.
[50] L. Zhong and K. Xu, Inequalities between vertex-degree-based topological indices, MATCH Commun. Math. Comput. Chem. 71 (2014), 627–642.
[51] H. Hosoya, Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons, Bulletin of the Chemical Society of Japan 44 (1971), 2332–2339.
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