Mathematics and Computer Science
Volume 1, Issue 4, November 2016, Pages: 101-107
Received: Sep. 3, 2016;
Accepted: Nov. 16, 2016;
Published: Dec. 20, 2016
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Mohammad Javad Nikmehr, Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran
Najmeh Soleimani, Young Researchers and Elite Club, Karj Branch, Islamic Azad University, Karaj, Iran
For a commutative ring
, the total graph of
which denoted by
, is a graph with all elements of R
as vertices, and two distinct vertices
are adjacent if and only if
denotes the set of zero-divisors of R
. In an earlier study, we computed Wiener, hyper-Wiener, reverse Wiener, Randic ́, Zagreb,
indices of zero-divisor graph. In this study, some computer programs are prepared to calculate the zero-divisors and adjacency matrix of the given graph which, apply these programs to compute the energy and first edge-Wiener, sum-connectivity, harmonic, augmented Zagreb and hyper-Zagreb indices.
Anderson, D. F. and Badawi, A. "The total graph of a commutative ring." Journal of Algebra. Vol. 320, 2008, pp. 2706-2719.
Dos ̌lic ́, T. and Saheli, M. "Augmented eccentric connectivity index of single defect nanocones." Journal of mathematical nanoscience. Vol. 1, No. 1, 2011, pp. 25-31.
Heydari, A. and Taeri, B. "Wiener and Schultz indices of TUC_4 C_8 (S) nanotubes." MATCH Communications in Mathematical and in Computer Chemistry. Vol. 57, 2007, pp. 665-676.
Nikmehr, M. J., Soleimani, N. and Tavallaee, H. A. "Computing some topological indices of carbon nanotubes." Proceedings of the Institute of Applied Mathematics. Vol. 4, No. 1, 2015, pp. 20-25.
Soleimani, N., Nikmehr, M. J. and Tavallaee, H. A. "Theoretical study of nanostructures using topological indices." Studia Universitatis Babes-Bolyai, Chemia. Vol. 59, No. 4, 2014, pp. 139-148.
Soleimani, N., Nikmehr, M. J. and Tavallaee, H. A. "Computation of the different topological indices of nanostructures." Journal of the national science foundation of Sri Lanka. Vol. 43, No. 2, 2015, pp. 127-133.
Soleimani, N., Mohseni, E. and Maleki, N. "Connectivity indices of some famous dendrimers." Journal of Chemical and Pharmaceutical Research. Vol. 8, No. 8, 2016, pp. 229-235.
Nikmehr, M. J., Heidarzadeh, L. and Soleimani, N. "Calculating different topological indices of total graph of Z_n." Studia Scientiarum Mathematicarum Hungarica. Vol. 51, No. 1, 2014, pp. 133-140.
Gutman, I. and Polansky, O. E. "Mathematical Concepts in Organic Chemistry." Springer-Verlag, Berlin. 1986.
Gutman, I. "The energy of a graph: old and new results." in Algebraic Combinatorics and Applications, Betten A., Kohner, R. Laue A. & Wassermann A., eds., Springer, Berlin. 2001, pp. 196-211.
Wiener, H. "Structural determination of the paraffin boiling points." Journal of the American Chemical Society. Vol. 69, 1947, pp. 17-20
Iranmanesh, A., Gutman, I., Khormali, O. and Mahmiani, A. "The edge versions of the Wiener index." MATCH Communications in Mathematical and in Computer Chemistry. Vol. 61, 2009, pp. 663-672.
Zhou, B. and Trinajstić, N. "On a novel connectivity index." Journal of Mathematical Chemistry. Vol. 46, 2009, pp. 1252-1270.
Fajtlowicz, S. "On conjectures of Graffiti-II." Congr. Numer. Vol. 60, 1987, pp. 187-197.
Furtula, B., Graovac, A. and Vukičević, D. "Augmented Zagreb index." Journal of Mathematical Chemistry. Vol. 48, 2010, pp. 370-380.
Shirdel, G. H., Rezapour, H. and Sayadi, A. M. "The Hyper-Zagreb index of graph operations." Iranian Journal of Mathematical Chemistry. Vol. 4, No. 2, 2013, pp. 213-220.
Maimani, H. R., Wickham, C. and Yassemi, S. "Rings whose total graphs have genus at most one." Rocky Mountain Journal of Mathematics. Vol. 42, No. 5, 2012, pp. 1551-1560.
Caporossi, G., Cvetković, D., Gutman, I and Hansen, P. "Variable neighborhood search for extremal graphs. 2. Finding graphs with external energy." Journal of Chemical Information and Computer Sciences. Vol. 39, 1999, pp. 984-996.