Scientific Research

An Academic Publisher

Some Results on Generalized Degree Distance

In [1], Hamzeh, Iranmanesh and Hossein-Zadeh and M. V. Diudea recently
introduced the generalized degree distance of graphs. In this paper, we
present explicit formulas for this new graph invariant of the Cartesian product,
composition, join, disjunction and symmetric difference of graphs and introduce
generalized and modified generalized degree distance polynomials of graphs,
such that their first derivatives at *x* = 1 are
respectively, equal to the generalized degree distance and the modified
generalized degree distance. These polynomials are related to Wiener-type invariant polynomial of graphs.

KEYWORDS

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

*Open Journal of Discrete Mathematics*, Vol. 3 No. 3, 2013, pp. 143-150. doi: 10.4236/ojdm.2013.33026.

[1] | A. Hamzeh, A. Iranmanesh, S. Hossein-Zadeh and M. V. Diudea, “Generalized Degree Distance of Trees, Unicyclic and Bicyclic Graphs,” Studia Ubb Chemia, LVII, Vol. 4, 2012, pp. 73-85. |

[2] | M. V. Diudea, I. Gutman and L. Jantschi, “Molecular Topology,” Nova Science, Huntington, 2001. |

[3] | W. Imrich and S. Klavzar, “Product Graphs: Structure and Recognition,” John Wiley & Sons, New York, 2000. |

[4] | I. Gutman and N. Trinajstic, “Graph Theory and Molecular Orbitals. Total π-Electron Energy of Alternant Hydrocarbons,” Chemical Physics Letters, Vol. 17, No. 4, 1972, pp. 535-538. doi:10.1016/0009-2614(72)85099-1 |

[5] | S. Nikolic, G. Kovacevic, A. Milicevic and N. Trinajstic, “The Zagreb Indices 30 Years after,” Croatica Chemica Acta, Vol. 76, No. 2, 2003, pp. 113-124. |

[6] | T. Doslic, “Vertex-Weighted Wiener Polynomials for Composite Graphs,” Ars Mathematica Contemporanea, Vol. 1, No. 1, 2008, pp. 66-80. |

[7] | I. Gutman, “A Property of the Wiener Number and Its Modifications,” Indian Journal of Chemistry, Vol. 36A, No. 2, 1997, pp. 128-132. |

[8] | I. Gutman, A. A. Dobrynin, S. Klavzar and L. Pavlovic, “Wiener-Type Invariants of Trees and Their Relation,” Bulletin of the Institute of Combinatorics and Its Applications, Vol. 40, No. 1, 2004, pp. 23-30. |

[9] | Y. Alizadeh, A. Iranmanesh and T. Doslic, “Additively Weighted Harary Index of Some Composite Graphs,” Discrete Mathematics, Vol. 313, No. 1, 2013, pp. 26-34. doi:10.1016/j.disc.2012.09.011 |

[10] | A. A. Dobrynin and A. A. Kochetova, “Degree Distance of a Graph: A Degree Analogue of the Wiener Index,” Journal of Chemical Information and Computer Sciences, Vol. 34, No. 5, 1994, pp. 1082-1086. doi:10.1021/ci00021a008 |

[11] | I. Gutman, “Selected Properties of the Schultz Molecular Topological Index,” Journal of Chemical Information and Computer Sciences, Vol. 34, No. 5, 1994, pp. 1087-1089. doi:10.1021/ci00021a009 |

[12] | I. Gutman, and S. Klavzar, “Wiener Number of Vertex-Weighted Graphs and a Chemical Applications,” Discrete Applied Mathematics, Vol. 80, No. 1, 1997, pp. 73-81. doi:10.1016/S0166-218X(97)00070-X |

[13] | H. Hua and S. Zhang, “On the Reciprocal Degree Distance of Graphs,” Discrete Applied Mathematics, Vol. 160, No. 7-8, 2012, pp. 1152-1163. doi:10.1016/j.dam.2011.11.032 |

[14] | S. Hossein-Zadeh, A. Hamzeh and A. R. Ashrafi, “Extremal Properties of Zagreb Coindices and Degree Distance of Graphs,” Miskolc Mathematical Notes, Vol. 11, No. 2, 2010, pp. 129-137. |

[15] | I. Tomescu, “Unicyclic and Bicyclic Graphs Having Minimum Degree Distance,” Discrete Applied Mathematics, Vol. 156, No. 1, 2008, pp. 125-130. doi:10.1016/j.dam.2007.09.010 |

[16] | I. Tomescu, “Some Extremal Properties of the Degree Distance of a Graph,” Discrete Applied Mathematics, Vol. 98, No. 1-2, 1999, pp. 159-163. doi:10.1016/S0166-218X(99)00117-1 |

[17] | A. Graovac and T. Pisanski, “On the Wiener Index of a Graph,” Journal of Mathematical Chemistry, Vol. 8, No. 1, 1991, pp. 53-62. doi:10.1007/BF01166923 |

[18] | B. E. Sagan, Y.-N. Yeh and P. Zhang, “The Wiener Polynomial of a Graph,” International Journal of Quantum Chemistry, Vol. 60, No. 5, 1996, pp. 959-969. doi:10.1002/(SICI)1097-461X(1996)60:5<959::AID-QUA2>3.0.CO;2-W |

[19] | S. Klavzar, A. Rajapakse and I. Gutman, “The Szeged and the Wiener Index of Graphs,” Applied Mathematics Letters, Vol. 9, No. 5, 1996, pp. 45-49. doi:10.1016/0893-9659(96)00071-7 |

[20] | M. H. Khalifeh, H. Yousefi-Azari and A. R. Ashrafi, “The Hyper-Wiener Index of Graph Operations,” Computers & Mathematics with Applications, Vol. 56, No. 5, 2008, pp. 1402-1407. doi:10.1016/j.camwa.2008.03.003 |

[21] | M. Eliasi and A. Iranmanesh, “The Hyper-Wiener Index of the Generalized Hierarchical Product of Graphs,” Discrete Applied Mathematics, Vol. 159, No. 8, 2011, pp. 866-871. doi:10.1016/j.dam.2010.12.020 |

[22] | M. Eliasi and B. Taeri, “Schultz Polynomials of Composite Graphs,” Applicable Analysis and Discrete Mathematics, Vol. 2, No. 2, 2008, pp. 285-296. doi:10.2298/AADM0802285E |

[23] | S. Hossein-Zadeh, A. Hamzeh and A. R. Ashrafi, “Wiener-Type Invariants of Some Graph Operations,” Filomat, Vol. 29, No. 3, 2009, pp. 103-113. doi:10.2298/FIL0903103H |

[24] | M. H. Khalifeh, H. Yousefi-Azari and A. R. Ashrafi, “The First and Second Zagreb Indices of Some Graph Operations,” Discrete Applied Mathematics, Vol. 157, No. 4, 2009, pp. 804-811. doi:10.1016/j.dam.2008.06.015 |

[25] | F. Harary, “Graph Theory,” Addison-Wesley, Reading, 1969. |

[26] | N. Trinajstic, “Chemical Graph Theory,” CRC Press, Boca Raton, 1992. |

Copyright © 2019 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.