^{2}Thiruvalluvar College of Engineering and Technology

Abstract

The reciprocal degree distance (RDD), defined for a connected graph $G$ as vertex-degree-weighted sum of the reciprocal distances, that is, $RDD(G) =\sum\limits_{u,v\in V(G)}\frac{d_G(u) + d_G(v)}{d_G(u,v)}.$ The reciprocal degree distance is a weight version of the Harary index, just as the degree distance is a weight version of the Wiener index. In this paper, we present exact formulae for the reciprocal degree distance of join, tensor product, strong product and wreath product of graphs in terms of other graph invariants including the degree distance, Harary index, the first Zagreb index and first Zagreb coindex. Finally, we apply some of our results to compute the reciprocal degree distance of fan graph, wheel graph, open fence and closed fence graphs.

N. Alon and E. Lubetzky (2007). Independent set in tensor graph powers. J. Graph Theory. 54, 73-87

2

A. M. Assaf (1990). Modified group divisible designs. Ars Combin.. 29, 13-20

3

B. Bresar, W. Imrich, S. Klavv{z}ar and B. Zmazek (2005). Hypercubes as direct products. SIAM J. Discrete Math.. 18, 778-786

4

K. C. Das, B. Zhou and N. Trinajsti'c (2009). Bounds on Harary index. J. Math. Chem.. 46, 1377-1393

5

A. A. Dobrynin and A. A. Kochetova (1994). Degree distance of a graph: a degree analogue of the Wiener index. J. Chem. Inf. Comput. Sci.. 34, 1082-1086

6

J. Devillers, A. T. Balaban and Eds. (1999). Topological indices and related descriptors in QSAR and QSPR. Gordon and Breach, Amsterdam, The Netherlands.

7

L. Feng and A. Ilic (2010). Zagreb, Harary and hyper-Wiener indices of graphs with a given matching number. Appl. Math. Lett.. 23, 943-948

8

I. Gutman (1994). Selected properties of the Schultz molecular topological index. J. Chem. Inf. Comput. Sci.. 34, 1087-1089

9

I. Gutman and O. E. Polansky (1986). Mathematical Concepts in Organic Chemistry. Springer-Verlag, Berlin.

10

M. Hoji, Z. Luo and E. Vumar (2010). Wiener and vertex PI indices of kronecker products of graphs. Discrete Appl. Math.. 158, 1848-1855

11

H. Hua and S. Zhang (2012). On the reciprocal degree distance of graphs. Discrete Appl. Math.. 160, 1152-1163

12

W. Imrich and S. Klavzar (2000). Product graphs: Structure and Recognition. John Wiley, New York.

13

O. Ivanciuc and T. S. Balaban (1993). Reciprocal distance matrix, related local vertex invariants and topological indices. J. Math. Chem.. 12, 309-318

14

M. H. Khalifeh, H. Youseri-Azari and A. R. Ashrafi (2008). Vertex and edge PI indices of Cartesian product of graphs. Discrete Appl. Math.. 156, 1780-1789

15

B. Lucic, A. Milicevic, S. Nikolic and N. Trinajstic (2002). Harary index-twelve years later. Croat. Chem. Acta. 75, 847-868

16

A. Mamut and E. Vumar (2008). Vertex vulnerability parameters of Kronecker products of complete graphs. Inform. Process. Lett.. 106, 258-262

17

K. Pattabiraman and P. Paulraja (2012). On some topological indices of the tensor product of graphs. Discrete Appl. Math.. 160, 267-279

18

K. Pattabiraman and P. Paulraja (2012). Wiener and vertex PI indices of the strong product of graphs. Discuss. Math. Graph Thoery. 32, 749-769

19

K. Pattabiraman and P. Paulraja (2011). Wiener index of the tensor product of a path and a cycle. Discuss. Math. Graph Thoery. 31, 737-751

20

D. Plavsic, S. Nikolic, N. Trinajstic and Z. Mihalic (1993). On the Harary index for the characterization of chemical graphs. J. Math. Chem.. 12, 235-250

21

K. Xu and K. C. Das (2011). On Harary index of graphs. Discrete. Appl. Math.. 159, 1631-1640

22

H. Yousefi-Azari, M. H. Khalifeh and A. R. Ashrafi (2011). Calculating the edge Wiener and edge Szeged indices of graphs. J. Comput. Appl. Math.. 235, 4866-4870

23

B. Zhou, Z. Du and N. Trinajstic (2008). Harary index of landscape graphs. Int. J. Chem. Model.. 1, 35-44

24

B. Zhou, X. Cai and N. Trinajstic (2008). On the Harary index. J. Math. Chem.. 44, 611-618