Structural and spectral properties of corona graphs

Product graphs have been gainfully used in literature to generate mathematical models of complex networks which inherit properties of real networks. Realizing the duplication phenomena imbibed in the definition of corona product of two graphs, we define Corona graphs. Given a small simple connected graph which we call seed graph, Corona graphs are defined by taking corona product of a seed graph iteratively. We show that the cumulative degree distribution of Corona graphs decay exponentially when the seed graph is regular and cumulative betweenness distribution follows power law when seed graph is a clique. We determine explicit formulae of eigenvalues, Laplacian eigenvalues and signless Laplacian eigenvalues of Corona graphs when the seed graph is regular. Computable expressions of eigenvalues and signless Laplacian eigenvalues of Corona graphs are also obtained when the seed graph is a star graph.

[1]  Alessandro Vespignani,et al.  Large-scale topological and dynamical properties of the Internet. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  D. Fell,et al.  The small world inside large metabolic networks , 2000, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[3]  S. Rachev.  Handbook of heavy tailed distributions in finance , 2003 .

[4]  Xintao Wu,et al.  Analysis of Spectral Space Properties of Directed Graphs Using Matrix Perturbation Theory with Application in Graph Partition , 2015, 2015 IEEE International Conference on Data Mining.

[5]  B. Kahng,et al.  Geometric fractal growth model for scale-free networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  F. Harary,et al.  On the corona of two graphs , 1970 .

[7]  Kwang-Il Goh,et al.  Goh et al. Reply , 2003 .

[8]  Kwang-Il Goh,et al.  Packet transport along the shortest pathways in scale-free networks , 2004 .

[9]  Cam McLeman,et al.  Spectra of coronae , 2011, 1111.1200.

[10]  R. Cooke,et al.  Fat-Tailed Distributions: Data, Diagnostics and Dependence , 2014 .

[11]  A. Jamakovic,et al.  Characterization of complex networks: Application to robustness analysis , 2008 .

[12]  Gábor Csárdi,et al.  The igraph software package for complex network research , 2006 .

[13]  Willem H. Haemers,et al.  Enumeration of cospectral graphs , 2004, Eur. J. Comb..

[14]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[15]  M. Crovella,et al.  Estimating the Heavy Tail Index from Scaling Properties , 1999 .

[16]  Beom Jun Kim,et al.  Vertex overload breakdown in evolving networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  R. Nickalls.  A new approach to solving the cubic: Cardan’s solution revealed , 1993, The Mathematical Gazette.

[18]  Shuigeng Zhou,et al.  Structural and spectral properties of a family of deterministic recursive trees: rigorous solutions , 2008, 0812.1456.

[19]  B. K. Sarma,et al.  The Spectrum of the Corona of Two Graphs , 2007, SIAM J. Discret. Math..

[20]  D. Cvetkovic,et al.  Signless Laplacians of finite graphs , 2007 .

[21]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[22]  Piet Van Mieghem,et al.  On the Robustness of Complex Networks by Using the Algebraic Connectivity , 2008, Networking.

[23]  S. Barik,et al.  UNICYCLIC GRAPHS WITH THE STRONG RECIPROCAL EIGENVALUE PROPERTY , 2008 .

[24]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[25]  Petter Holme.  Edge overload breakdown in evolving networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Paul J. Laurienti,et al.  Universal fractal scaling of self-organized networks , 2010 .

[27]  Lucye Guilbeau.  The History of the Solution of the Cubic Equation , 1930 .

[28]  Matthew Roughan,et al.  Generalized graph products for network design and analysis , 2011, 2011 19th IEEE International Conference on Network Protocols.

[29]  Christos Faloutsos,et al.  R-MAT: A Recursive Model for Graph Mining , 2004, SDM.

[30]  Shu-Yu Cui,et al.  The spectrum and the signless Laplacian spectrum of coronae , 2012 .

[31]  D. Cvetkovic,et al.  Towards a spectral theory of graphs based on the signless Laplacian, I , 2009 .

[32]  V. Latora,et al.  Complex networks: Structure and dynamics , 2006 .

[33]  Cvetkovicccc Dragos,et al.  Towards a spectral theory of graphs based on the signless Laplacian, III , 2010 .

[34]  I. Ispolatov,et al.  Duplication-divergence model of protein interaction network. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  Peter J. Cameron,et al.  Graphs and matrices , 2004 .

[36]  R. Solé,et al.  Optimization in Complex Networks , 2001, cond-mat/0111222.

[37]  Xiaogang Liu,et al.  Spectra of subdivision-vertex and subdivision-edge neighbourhood coronae , 2012, 1212.0851.

[38]  Chonghui Guo,et al.  A deterministic small-world network created by edge iterations , 2006 .

[39]  J. Kurths,et al.  Hierarchical synchronization in complex networks with heterogeneous degrees. , 2006, Chaos.

[40]  Christos Faloutsos,et al.  Kronecker Graphs: An Approach to Modeling Networks , 2008, J. Mach. Learn. Res..

[41]  Dragoš Cvetković,et al.  Towards a spectral theory of graphs based on the signless Laplacian, III , 2010 .

[42]  K. Goh,et al.  Universal behavior of load distribution in scale-free networks. , 2001, Physical review letters.