Wiener, Hyper Wiener and Detour Index of Pseudoregular Graphs

Authors

  • S. Kavithaa R&D Centre, Bharatiyar University, Coimbatore, Tamilnadu, India; Department of Mathematics, Thanthai Hans Roever College, Perambalur 621212, Tamilnadu, India
  • V. Kaladevi P.G. & Research Department of Mathematics, Bishop Heber College, Trichy 17, Tamilnadu, India

DOI:

https://doi.org/10.26713/jims.v9i3.937

Keywords:

Wiener index, Hyper wiener index, Pseudo-regular graphs.

Abstract

The field of graph theory is rich in its theoretical and application area. Drugs and other chemical compounds are often modeled as polygonal shape, where each vertex represents an atom of the molecule and covalent bonds between atoms are represented by edges between the corresponding vertices. This polygonal shape derived from a chemical compound is often called its modecular graph and can be a path, a tree or in general graph. An indicator defined over this molecular graph, the Wiener index \(W(G)\) is defined as \(W(G)=\sum\limits _{u\ne v}d(u,v) \), where the sum is taken through all unordered pairs of vertices of \(G\).  Another indicator is the Hyper Wiener index
\(WW(G)\) is defined as \(WW(G)=\frac{1}{2} \sum\limits _{u\ne v}\left(d(u,v)+d^{2} (u,v)\right)\), where \(d^{2}(u,v)=d(u,v)^{2}\), and the Detour index \(D(G)\) is defined as \(D(G)=\sum\limits _{u\ne v}D(u,v)\), where \(D(u,v)\) denotes the longest distance from \(u\) to \(v\) in \(G\). In this paper, we computed the Wiener, Hyper Wiener, Detour index for a special graph namely, Pseudo-regular graph.

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References

H. Wiener, Structural determination of Paraffin boiling points, J. Amer. Chem. Soc. 69 (1947), 17 – 20.

I. Gutman and I.G. Zenkevich, Wiener index and vibrational energy, Z. Naturforsch. 57 (2002), 824 – 828.

B. Elenbogen and J.F. Fink, Distance distribution for graphs modeling computer networks, Discrete Applied Math. 155 (2007), 2612 – 2624.

D.J. Klein, I. Lukovits and I. Gutman, On the definition of the hyper-wiener index for cycle containing structures, J. Chem. Inf. Computer Sci. 35 (1995), 50 – 52.

I. Gutman, Relation between Hyper-Wiener and wiener index, Chem. Phys. Letter 364 (2002), 352 – 356.

G.G. Cash, Polynomial expressions for the Hyper wiener index of extended hydrocarbon networks, Comput. Chem. 25 (2001), 577 – 582.

A.A. Dobrynin, R. Entringer and I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math. 66 (3) (2001), 211 – 249.

R.C. Entringer, D.E. Jackson and D.A. Synder, Distance in graphs, Czechoslovak Math. J. 26 (1976), 283 – 296.

G. Chartrand, H. Escuadro and P. Zhang, Detour distance in graphs, J. Combin. Math. Combin. Comput. 53 (2005), 75 – 94.

G. Chartrand, G.L. Johns and P. Zhang, On the detour number and geodetic number of a graph, Ars Combin. 72 (2004), 3 – 15.

A.P. Santhakumaran and S.V. Ullas Chandran, The vertex Detour Hull number of a graph, Discussiones Mathematicae Graph Theory 32 (2012), 321 – 330.

A. Joshi and B. Babujee, J. Wiener Polynomial for graph with cycles, ICMCS, 119 – 225 (2008).

V. Kaladevi and P. Backialakshmi, Detour distance polynomial of double star graph and Cartesian product of P2 and Cn, Antartica Journal of Mathematics 8 (2011), 399.

V. Kaladevi and P. Selvarani, Three polynomials in one matrix, Mathematical Sciences International Research Journal 1 (1).

D.S. Cao, Bounds on eigen values and Chromatic numbers, Linear Algebra Appl. 270 (1998), 1 – 13. [16] M.V. Diudea (ed.), Nanostructures, Novel Architecture, Nova, New York (2006).

M.V. Diudea, Nanomolecules and Nanostructures, Univ. Kragujevac (2010).

V. Kaladevi and S. Kavithaa, Fifteen Reverse topological indices of a graph in a single distance matrix, Jamal Academic Research Journal 7(1,2) (2013-2014), 303, 1 – 9.

V. Kaladevi and S. Kavithaa, On varieties of reverse Wiener like indices of a graph, International Journal of Fuzzy mathematics Archive 4 (1), 2320 – 3250, 37 – 46.

A.T. Balaban, D. Mills, O. Ivanciuc and S.C. Basak, Reverse Wiener indices, Croat. Chem. Acta 73 (2000), 923 – 941.

G. Rucker and C. Rucker, Symmetry aided computation of the detour matrix and the detour index, J. Chem Inf. Computer Sci. 38 (1998), 710 – 714.

I. Lukovits and M. Razinger, On calculation of the Detour index, J. Chem Inf. Computer Sci. 38 (1997), 283 – 286.

S. Nikolic, N. Trinajstic and Z. Mihalic, The Detour Matrix and the Detour index, in: J. Devillers and A.T. Balaban (eds.), Topological Indices and Related Descriptors in QSAR & QSPR Gordon and Breach, Amsterdam, 279 – 306 (1999).

M.L. Aimeiyu and F. Tian, On the spectral radius of graphs, Linear Algebra and its Applications 387 (2004), 41 – 49.

T. Reti, I. Gutman and D. Vukicovic, On Zagreb indices of pseudo-regular graphs, Journal of Mathematical Nanoscience 1 (1) (2011), 1 – 12.

B. Zhou and X.C. Cai, On Detour index, Match Communication in Mathematical and in Computer Chemistry 63 (2010), 199 – 210.

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Published

2017-10-31
CITATION

How to Cite

Kavithaa, S., & Kaladevi, V. (2017). Wiener, Hyper Wiener and Detour Index of Pseudoregular Graphs. Journal of Informatics and Mathematical Sciences, 9(3), 751–763. https://doi.org/10.26713/jims.v9i3.937

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Research Articles