Total Distance, Wiener Index and Opportunity Index in Wreath Products of Star Graphs

In the last decades much attention has turned towards centrality measures on graphs. The Wiener index and the total distance are key tools to investigate the median vertices, the distance-balanced property and the opportunity index of a graph. This interest has recently been addressed to graphs obtained via classical graph products like the Cartesian, the direct, the strong and the lexicographic product. We extend this study to a relatively new graph product, that is, the wreath product. In this paper, we compute the total distance for the vertices of an arbitrary wreath product graph G oH in terms of the total distances in H and of some distance-based indices of G. We explicitly compute these indices for the star graph Sn, providing a closed formula for the total distances in Sn o H when H is complete or a star. As a consequence, we obtain the Wiener index of these graphs, we characterize the median and the central vertices, and finally we give an upper and a lower bound for the opportunity index of Sn o Sm in terms of tail conditional expectations of an associated binomial distribution. Mathematics Subject Classifications: 05C12, 05C57, 05C76

[1]  N. Trinajstic.  Chemical Graph Theory , 1992 .

[2]  Wolfgang Woess,et al.  A Note on the Norms of Transition Operators on Lamplighter Graphs and Groups , 2005, Int. J. Algebra Comput..

[3]  Keiichi Handa.  Bipartitie Graphs with Balanced (a, b)-Partitions , 1999, Ars Comb..

[4]  Sandi Klavzar,et al.  On distance-balanced graphs , 2010, Eur. J. Comb..

[5]  Alfredo Donno.  Spectrum, distance spectrum, and Wiener index of wreath products of complete graphs , 2017, Ars Math. Contemp..

[6]  S. L. HAKIMIt.  AN ALGORITHMIC APPROACH TO NETWORK LOCATION PROBLEMS. , 1979 .

[7]  Kannan Balakrishnan,et al.  Strongly distance-balanced graphs and graph products , 2009, Eur. J. Comb..

[8]  Wreath product of matrices , 2015, 1507.02609.

[9]  Alfredo Donno.  Generalized Wreath Products of Graphs and Groups , 2015, Graphs Comb..

[10]  K. Reid.  Balance vertices in trees , 1999, Networks.

[11]  J. D. P. Meldrum.  Wreath Products of Groups and Semigroups , 1995 .

[12]  H. Oser.  An Average Distance , 1975 .

[13]  Diane Vizine-Goetz,et al.  Spectrum , 2001 .

[14]  Aleksander Vesel,et al.  Equal opportunity networks, distance-balanced graphs, and Wiener game , 2014, Discret. Optim..

[15]  Dragan Marusic,et al.  Distance-balanced graphs: Symmetry conditions , 2006, Discret. Math..

[16]  Ehsan Pourhadi,et al.  Graph Theory in Distribution and Transportation Problems and the Connection to Distance-Balanced Graphs , 2016 .

[17]  Francesco Belardo,et al.  Wreath product of a complete graph with a cyclic graph: Topological indices and spectrum , 2018, Appl. Math. Comput..

[18]  Peter Dankelmann,et al.  Average distance in weighted graphs , 2012, Discret. Math..

[19]  Patti Frazer Lock,et al.  A Survey of Graphs Hamiltonian-Connected from a Vertex , 2009 .

[20]  H. Wiener.  Structural determination of paraffin boiling points. , 1947, Journal of the American Chemical Society.

[21]  Cheryl E. Praeger,et al.  Generalized Wreath Products of Permutation Groups , 1983 .

[22]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[23]  K. B. Reid.  Centrality Measures in Trees , 2010 .

[24]  Sergio Cabello,et al.  The Complexity of Obtaining a Distance-Balanced Graph , 2011, Electron. J. Comb..

[25]  Sheldon M. Ross,et al.  Introduction to probability models , 1975 .