Harary Index of Product Graphs

Abstract The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. In this paper, the exact formulae for the Harary indices of tensor product G × Km0,m1,...,mr−1 and the strong product G⊠Km0,m1,...,mr−1 , where Km0,m1,...,mr−1 is the complete multipartite graph with partite sets of sizes m0,m1, . . . ,mr−1 are obtained. Also upper bounds for the Harary indices of tensor and strong products of graphs are estabilished. Finally, the exact formula for the Harary index of the wreath product G ○ G′ is obtained.

[1]  A. Balaban,et al.  Topological Indices and Related Descriptors in QSAR and QSPR , 2003 .

[2]  Lihua Feng,et al.  Zagreb, Harary and hyper-Wiener indices of graphs with a given matching number , 2010, Appl. Math. Lett..

[3]  Kinkar Chandra Das,et al.  On Harary index of graphs , 2011, Discret. Appl. Math..

[4]  Ali Reza Ashrafi,et al.  Calculating the edge Wiener and edge Szeged indices of graphs , 2011, J. Comput. Appl. Math..

[5]  Noga Alon,et al.  Independent sets in tensor graph powers , 2007, J. Graph Theory.

[6]  Roberto Todeschini,et al.  Handbook of Molecular Descriptors , 2002 .

[7]  P. Paulraja,et al.  Wiener and vertex PI indices of the strong product of graphs , 2012, Discuss. Math. Graph Theory.

[8]  Ali Reza Ashrafi,et al.  Vertex and edge PI indices of Cartesian product graphs , 2008, Discret. Appl. Math..

[9]  Aygul Mamut,et al.  Vertex vulnerability parameters of Kronecker products of complete graphs , 2008, Inf. Process. Lett..

[10]  Bo Zhou,et al.  Bounds on Harary index , 2009 .

[11]  N. Trinajstic,et al.  On the Harary index for the characterization of chemical graphs , 1993 .

[12]  Ivan Gutman,et al.  A PROPERTY OF THE WIENER NUMBER AND ITS MODIFICATIONS , 1997 .

[13]  P. Paulraja,et al.  Wiener index of the tensor product of a path and a cycle , 2011, Discuss. Math. Graph Theory.

[14]  N. Alon,et al.  Independent sets in tensor graph powers , 2007 .

[15]  Ovidiu Ivanciuc,et al.  Design of topological indices. Part 4. Reciprocal distance matrix, related local vertex invariants and topological indices , 1993 .

[16]  Nenad Trinajstić,et al.  Harary Index - Twelve Years Later* , 2002 .

[17]  I. Gutman,et al.  Mathematical Concepts in Organic Chemistry , 1986 .

[18]  Mircea V. Diudea,et al.  Indices of Reciprocal Properties or Harary Indices , 1997, J. Chem. Inf. Comput. Sci..

[19]  Bo Zhou,et al.  On Harary index , 2008 .

[20]  Elkin Vumar,et al.  Wiener and vertex PI indices of Kronecker products of graphs , 2010, Discret. Appl. Math..

[21]  S C Basak,et al.  Distance Indices and Their Hyper-Counterparts: Intercorrelation and Use in the Structure-Property Modeling , 2001, SAR and QSAR in environmental research.

[22]  R. Balakrishnan,et al.  A textbook of graph theory , 1999 .

[23]  P. Seybold,et al.  Molecular modeling of the physical properties of the alkanes , 1988 .

[24]  Wilfried Imrich,et al.  Hypercubes As Direct Products , 2005, SIAM J. Discret. Math..