Some results on vertex-edge Wiener polynomials and indices of graphs

Subject Areas : Statistics

M. azari ¹

1 - Assistant Professor, Department of Mathematics, Kazerun Branch, Islamic Azad University, P. O. Box:73135-168, Kazerun, Iran

Received: 2017-08-02 Accepted : 2018-09-01 Published : 2018-10-23

Keywords: شاخص توپولوژیک, حاصل‌ضرب دکارتی گراف‌ها, نانولوله, نانوچنبره,

Abstract :

The vertex-edge Wiener polynomials of a simple connected graph are defined based on the distances between vertices and edges of that graph. The first derivative of these polynomials at one are called the vertex-edge Wiener indices. In this paper, we express some basic properties of the first and second vertex-edge Wiener polynomials of simple connected graphs and compare the first and second vertex-edge Wiener indices of them with each other. Also, we compute these polynomials and indices for some well-known graphs. Then, we study the relation between the vertex-edge Wiener polynomials of Cartesian product of graphs with the Wiener polynomial and vertex-edge Wiener polynomials of the primary graphs and apply the results to compute the vertex-edge Wiener indices of Cartesian product of graphs. As applications of these results, we present exact formulas for computing the first and second vertex-edge Wiener indices of rectangular grids, C4-nanotubes, C4-nanotori, Hamming graph, and hypercubes.

References:

[1] N. Trinajstić, Chemical Graph Theory. CRC Press. Boca Raton (1992)

 

[2] R.   Todeschini, V. Consonni. Handbook of Molecular Descriptors. Wiley-VCH. Weinheim (2000)

 

[3] H. Wiener. Structural determination of paraffin boiling points. Journal of American Chemical Society. 69(1):17-20 (1947)

 

[4] Z. Yarahmadi, G. H. Fath-Tabar. The Wiener, Szeged, PI, vertex PI, the first and second Zagreb indices of N-branched phenylacetylenes dendrimers. MATCH Communications in Mathematical and in Computer Chemistry. 65(1):201-208 (2011)

 

[5] M. R. Darafsheh, M. H. Khalifeh. Calculation of the Wiener, Szeged, and PI indices of a certain nanostar dendrimer. Ars Combinatoria. 100: 289-298 (2011)

 

[6] A. R. Ashrafi. Wiener index of nanotubes, toroidal fullerenes and nanostars. In: F. Cataldo, A. Graovac, O. Ori (Eds.). The Mathematics and Topology of Fullerenes. Springer Netherlands. Dordrecht pp. 21-38 (2011)

 

[7] A. R. Ashrafi, Z. Mohammad-Abadi. On Wiener index of one-heptagonal nanocone. Current Nanoscience.8(1):180-185 (2012)

 

[8] Y. Alizadeh. Wiener index of SC₅C₇[p, q] nanotubes. Optoelectronics and Advanced Materials-Rapid Communications. 7(11):943-946 (2013)

 

[9] A. Heydari. On the Wiener and terminal Wiener index of generalized Bethe trees. MATCH Communications in Mathematical and in Computer Chemistry.  69:141-150 (2013)

 

[10] A. Nikseresht, Z. Sepasdar. On the Kirchhoff and the Wiener indices of graphs and block decomposition. Electronic Journal of Combinatorics. 21(1):# P1.25  (2014)

 

[11] H. Hosoya. On some counting polynomials in Chemistry. Discrete Applied Mathematics. 19:239-257  (1988)

 

[12] B. E. Sagan, Y. N. Yeh, P. Zhang. The Wiener polynomial of a graph. International Journal of Quantum Chemistry. 60(5):959-969 (1996)

 

[13] D. Stevanović. Hosoya polynomial of composite graphs. Discrete Mathematics. 235:237-244 (2001)

 

[14] M. V. Diudea. Hosoya polynomial in tori. MATCH Communications in Mathematical and in Computer Chemistry. 45:109-122 (2002)

 

[15] H. B. Walikar, H. S. Ramane, L. Sindagi, S. S. Shirakol, I. Gutman. Hosoya polynomial of thorn trees, rods, rings, and stars. Kragujevac Journal of Science. 28:47-56 (2006)

 

[16] M. Eliasi, A. Iranmanesh. Hosoya polynomial of hierarchical product of graphs. MATCH Communications in Mathematical and in Computer Chemistry. 69:111-119 (2013)

 

[17] M. H. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi, S. G. Wagner. Some new results on distance-based graph invariants. European Journal of Combinatorics. 30:1149-1163 (2009)

 

[18] M. Azari, A. Iranmanesh. Computation of the edge Wiener indices of the sum of graphs. Ars Combinatoria. 100:113-128 (2011)

 

[19] M. Azari, A. Iranmanesh, A. Tehranian. Two topological indices of three chemical structures.MATCH Communications in Mathematical and in Computer Chemistry. 69:69-86 (2013)

 

[20] M. Azari, A. Iranmanesh, A. Tehranian. Maximum and minimum polynomials of a composite graph. Australian Journal of Basic and Applied Sciences. 5(9):825-830 (2011)

 

[21] M. Azari, A. Iranmanesh. Computing Wiener-like topological invariants for some composite graphs and some nanotubes and nanotori. In: I. Gutman (Ed.). Topics in Chemical Graph Theory. Univ. Kragujevac. Kragujevac pp. 69-90 (2014)

 

[22] M. Azari, A. Iranmanesh. Clusters and various versions of Wiener-type invariants. Kragujevac Journal of Mathematics. 39(2):155-171 (2015)

 

[23] M. Azari. A note on vertex-edge Wiener indices of graphs. Iranian Journal of Mathematical Chemistry. 7(1):11-17 (2016)

 

[24] M. Azari, A. Iranmanesh. On the vertex-edge Wiener polynomials of the disjunctive product of graphs. Acta Universitatis Apulensis. 45(1):61-72 (2016)

[25] M. Azari, A. Iranmanesh. Joins, coronas and their vertex-edge Wiener polynomials. Tamkang Journal of Mathematics. 47(2):163-178 (2016)

 

[26] I. Gutman, N. Trinajstić. Graph theory and molecular orbitals. Total -electron energy of alternant hydrocarbons. Chemical Physics Letters. 17(4):535-538 (1972)