Abstract
Let G=(V,E) be a simple connected graph with vertex set V and edge set E. The Wiener index of G is defined by W(G)=∑{x,y}⊆V d(x,y), where d(x,y) is the length of the shortest path from x to y. The Szeged index of G is defined by Sz(G)=∑e=uv∈E n u (e|G)n v (e|G), where n u (e|G) (resp. n v (e|G)) is the number of vertices of G closer to u (resp. v) than v (resp. u). The Padmakar–Ivan index of G is defined by PI(G)=∑e=uv∈E[n eu (e|G)+n ev (e|G)], where n eu (e|G) (resp. n ev (e|G)) is the number of edges of G closer to u (resp. v) than v (resp. u). In this paper we find the above indices for various graphs using the group of automorphisms of G. This is an efficient method of finding these indices especially when the automorphism group of G has a few orbits on V or E. We also find the Wiener indices of a few graphs which frequently arise in mathematical chemistry using inductive methods.
Similar content being viewed by others
References
Darafsheh, M.R.: The Wiener, Szeged and PI index of the triangle graph (submitted for publication)
Deng, H., Chen, S.: PI indices of pericondensed benzenoid graphs. J. Math. Chem. 43(1), 19–25 (2008)
Dobrynin, A.A., Entringer, R., Gutman, I.: Wiener index of trees: Theory and applications. Acta Appl. Math. 66, 211–249 (2001)
Dobrynin, A.A., Gutman, I.: Congruence relations for the Szeged index of hexagonal chains. Univ. Beogr. Publ. Elektrotehn. Fak. Ser. Mat. 8, 106–113 (1997)
Dobrynin, A.A., Gutman, I., Klavzar, S., Zigert, P.: Wiener index of hexagonal systems. Acta Appl. Math. 72, 247–294 (2002)
Entringer, R.C., Jackson, D.E., Snyder, D.A.: Distance in graphs. Czechoslav. Math. J. 26, 283–296 (1976)
Gutman, I.: A formula for the Wiener number of trees and its extension to graphs containing cycles. Graph Theory Notes, NY 27, 9–15 (1994)
Gutman, I., Dobrynin, A.A.: The Szeged index—a success story. Graph Theory Notes, NY 34, 37–44 (1998)
Gutman, I., Klavzar, S., Mohr, B. (eds.): Fifty years of the Wiener index. MATCH Commun. Math. Comput. Chem. 35, 1–259 (1997)
Gutman, I., Yeh, Y.N., Lee, S.L., Chen, J.C.: Wiener numbers of dendrimers. MATCH Commun. Math. Comput. Chem. 30, 103–115 (1994)
Harary, F.: Graph Theory. Addison Wesley, Reading (1968)
Harary, F.: The automorphism group of a hypercube. J. Univers. Comput. Sci. 6, 136–138 (2000)
Hosoya, H.: Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons. Bull. Chem. Soc. Jpn. 4, 2332–2339 (1971)
Khadikar, P.V.: On a novel structural descriptor PI. Natl. Acad. Sci. Lett. 23, 113–118 (2000)
Khadikar, P.V., Deshpande, N.V., Kale, P.P., Dobrynin, A.A., Gutman, I.: The Szeged index and an analogy with the Wiener index. J. Chem. Inf. Comput. Sci. 35, 547–550 (1995)
Khadikar, P.V., Karmarkar, S., Agrawal, V.K.: Relationship and relative correction potential of the Wiener, Szeged and PI indexes. Natl. Acad. Sci. Lett. 23, 165–170 (2000)
Mansour, T., Schork, M.: The vertex PI index and Szeged index of bridge graphs. Discrete Appl. Math. 157(7), 1600–1606 (2009)
Prabhakara Rao, N., Laxmi Prasanna, A.: On the Wiener index of pentachains. Appl. Math. Sci. 2(49), 2443–2457 (2008)
Tang, Z., Deng, H.: The (n,n)-graphs with the first three extremal Wiener indices. J. Math. Chem. 43(1), 60–74 (2008)
Wiener, H.: Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17–20 (1947)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Darafsheh, M.R. Computation of Topological Indices of Some Graphs. Acta Appl Math 110, 1225–1235 (2010). https://doi.org/10.1007/s10440-009-9503-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-009-9503-8