Spectrum, distance spectrum, and Wiener index of wreath products of complete graphs

We describe the adjacency matrix and the distance matrix of the wreath product of two complete graphs, and we give an explicit computation of their spectra. As an application, we deduce the spectrum of the transition matrix of the Lamplighter random walk over a complete base graph, with a complete color graph. Finally, an explicit computation of the Wiener index is given.

[1]  Computing the Wiener index in Sierpinski carpet graphs , 2016, 1802.09840.

[2]  Avi Wigderson,et al.  Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

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

[4]  A. Erschler.  GENERALIZED WREATH PRODUCTS , 2006 .

[5]  I. Gutman,et al.  Wiener Index of Trees: Theory and Applications , 2001 .

[6]  A. Donno.  Replacement and zig-zag products, Cayley graphs and Lamplighter random walk , 2013 .

[7]  Daniele D'Angeli,et al.  Metric compactification of infinite Sierpiński carpet graphs , 2015, Discret. Math..

[8]  Bijan Taeri,et al.  Wiener index of some graph operations , 2012, Discret. Appl. Math..

[9]  Wilfried Imrich,et al.  Associative products of graphs , 1975 .

[10]  Gopalapillai Indulal,et al.  On the distance spectra of some graphs , 2008 .

[11]  Gert Sabidussi,et al.  Graph multiplication , 1959 .

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

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

[14]  Andrzej Żuk,et al.  The Lamplighter Group as a Group Generated by a 2-state Automaton, and its Spectrum , 2001 .

[15]  On the spectrum of lamplighter groups and percolation clusters , 2007, 0712.3135.

[16]  G. Tee.  Eigenvectors of block circulant and alternating circulant matrices , 2005 .

[17]  Gopalapillai Indulal,et al.  The distance spectrum and energy of the compositions of regular graphs , 2009, Appl. Math. Lett..

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

[19]  Noga Alon,et al.  Semi-direct product in groups and zig-zag product in graphs: connections and applications , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[20]  Daniele D'Angeli,et al.  Connectedness and Isomorphism Properties of the Zig‐Zag Product of Graphs , 2014, J. Graph Theory.

[21]  Willem H. Haemers,et al.  Spectra of Graphs , 2011 .

[22]  A. Donno,et al.  Isomorphism classification of infinite Sierpinski carpet graphs. , 2015, 1802.09839.

[23]  S. R. Simanca,et al.  On Circulant Matrices , 2012 .

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

[25]  Indulal Gopalapillai,et al.  Distance spectrum of graph compositions , 2009 .

[26]  F. Scarabotti,et al.  Harmonic analysis on a finite homogeneous space , 2007, math/0701533.

[27]  G. Sabidussi.  The composition of graphs , 1959 .

[28]  Łukasz Grabowski,et al.  On Turing dynamical systems and the Atiyah problem , 2010, Inventiones mathematicae.

[29]  Spectral Computations on Lamplighter Groups and Diestel-Leader Graphs , 2004, math/0405182.

[30]  Bijan Taeri,et al.  Four new sums of graphs and their Wiener indices , 2009, Discret. Appl. Math..

[31]  F. Scarabotti,et al.  Harmonic Analysis of Finite Lamplighter Random Walks , 2007, math/0701603.

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

[33]  A. Donno,et al.  Distances and isomorphisms in 4-regular circulant graphs , 2016 .

[34]  I. Gutman,et al.  Wiener Index of Hexagonal Systems , 2002 .

[35]  A. Hora,et al.  Distance-Regular Graphs , 2007 .