Entire Wiener index of graphs

Document Type : Original paper


1 Uludag University Faculty of Arts and Science Department of Mathematics Gorukle, 16059 Bursa/Turkey

2 Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia

3 Department of Mathematics, University of Aden, Yemen


Topological indices are graph invariants computed usually by means of the distances or degrees of vertices of a graph. In chemical graph theory, a molecule can be modeled by a graph by replacing atoms by the vertices and bonds by the edges of this graph. Topological graph indices have been successfully used in determining the structural properties and in predicting certain physicochemical properties of chemical compounds. Wiener index is the oldest topological index which can be used for analyzing intrinsic properties of a molecular structure in chemistry. The Wiener index of a graph $G$ is equal to the sum of distances between all pairs of vertices of $G$. Recently, the entire versions of several indices have been introduced and studied due to their applications. Here we introduce the entire Wiener index of a graph. Exact values of this index for trees and some graph families are obtained, some properties and bounds for the entire Wiener index are established. Exact values of this new index for subdivision and $k$-subdivision graphs and some graph operations are obtained.


Main Subjects

[1] M.B. Ahmadi, S.A. Hosseini, and P.S. Nowbandegani, On trees with minimal atom bond connectivity index, MATCH Commun. Math. Comput. Chem. 69 (2013), 559–563.
[2] M. Ascioglu and I.N. Cangul, Sigma index and forgotten index of the subdivision and r-subdivision graphs, Proceedings of the Jangjeon Mathematical Society 21 (2018), no. 3, 375–383.
[3] M. Azari, Multiplicative sum Zagreb index of chain and chain-cycle graphs, J. Inform. Optim. Sci. 40 (2019), no. 7, 1529–1541.
[4] F. Buckley, Mean distance in line graphs, Congr. Numer. 32 (1981), 153–162.
[5] A.A. Dobrynin, R. Entringer, and I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math. 66 (2001), no. 3, 211–249.
[6] A.A. Dobrynin, I. Gutman, Klavžar, and P. Zigert, ˇ Wiener index of hexagonal systems, Acta Appl. Math. 72 (2002), no. 3, 247–294.
[7] J.K. Doyle and J.E. Graver, Mean distance in a graph, Discrete Math. 17 (1977), no. 2, 147–154.
[8] R.C. Entringer, Distance in graphs: Trees, J. Comb. Math. Comb. Comput. 24 (1997), 65–84.
[9] I. Gutman, Selected properties of the Schultz molecular topological index, J. Chem. Inf. Coumput. Sci. 34 (1994), no. 5, 1087–1089.
[10] I. Gutman, Kragujevac trees and their energy, Ser. A: Appl. Math. Inform. Mech. 6 (2014), no. 2, 71–79.
[11] I. Gutman, A.M. Naji, and N.D. Soner, On leap Zagreb indices of graphs, Commun. Comb. Optim. 2 (2017), no. 2, 99–117.
[12] I. Gutman and O.E. Polansky, Mathematical Concepts in Organic Chemistry, Springer Science & Business Media, 2012.
[13] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total ϕelectron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), no. 4, 535–538.
[14] I. Gutman, Y.-N. Yeh, S.-L. Lee, and Y.-L. Luo, Some recent results in the theory of the Wiener number, Indian J. Chem. 32A (1993), 651–661.
[15] F. Harary, Graph Theory, Addison-Wesley, Reading Mass, 1969.
[16] A. Iranmanesh, I. Gutman, O. Khormali, and A. Mahmiani, The edge versions of the Wiener index, MATCH Commun. Math. Comput. Chem. 61 (2009), no. 3, 663–672.
[17] S. Nikolić, G. Kovačević, A. Miličević, and N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (2003), no. 2, 113–124.
[18] R. Skrekovski and I. Gutman,  Vertex version of the Wiener Theorem, MATCH Commun. Math. Comput. Chem. 72 (2014), no. 1, 295–300.
[19] M. Togan, A. Yurttas, and I.N. Cangul, Some formulae and inequalities on several Zagreb indices of r-subdivision graphs, Enlightments of Pure and Applied Mathematics (EPAM) 1 (2015), no. 1, 29–45.
[20] , Zagreb and multiplicative Zagreb indices of r-subdivision graphs of double graphs, Scientia Magna 12 (2017), no. 1, 115–119.
[21] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947), no. 1, 17–20.
[22] B. Wu, Wiener index of line graphs, MATCH Commun. Math. Comput. Chem. 64 (2010), no. 3, 699–706.
[23] Y.-N. Yeh and I. Gutman, On the sum of all distances in composite graphs, Discrete Math. 135 (1994), no. 1-3, 359–365.
[24] B. Zhou, Zagreb indices, MATCH Commun. Math. Comput. Chem. 52 (2004), no. 1, 13–118.