Applications of Laplacian spectra for n-prism networks

In this paper, the properties of the Laplacian matrices for the n-prism networks are investigated. We calculate the Laplacian spectra of n-prism graphs which are both planar and polyhedral. In particular, we derive the analytical expressions for the product and the sum of the reciprocals of all nonzero Laplacian eigenvalues. Moreover, these results are used to handle various problems that often arise in the study of networks including Kirchhoff index, global mean-first passage time, average path length and the number of spanning trees. These consequences improve and extend the earlier results. HighlightsWe propose the structure of n-prism networks.We calculate the Laplacian spectra of n-prism networks.We deduce expressions for product and sum of reciprocals of all nonzero Laplacian-eigenvalues.Kirchhoff index, GMFPT, average path length and the number of spanning trees are obtained.

[1]  Shuigeng Zhou,et al.  Recursive solutions for Laplacian spectra and eigenvectors of a class of growing treelike networks. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Jihad H. Asad,et al.  Resistance calculation of the decorated centered cubic networks: Applications of the Green's function , 2014 .

[3]  Bin Wu,et al.  The number of spanning trees in Apollonian networks , 2012, Discret. Appl. Math..

[4]  Jia-Bao Liu,et al.  Asymptotic incidence energy of lattices , 2015 .

[5]  Fangyue Chen,et al.  APPLICATIONS OF LAPLACIAN SPECTRA ON A 3-PRISM GRAPH , 2014 .

[6]  D. Dhar.  Theoretical studies of self-organized criticality , 2006 .

[7]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[8]  Sergey N. Dorogovtsev,et al.  Critical phenomena in complex networks , 2007, ArXiv.

[9]  T. S. Evans,et al.  Complex networks , 2004 .

[10]  Shlomo Havlin,et al.  Transport in weighted networks: partition into superhighways and roads. , 2006, Physical review letters.

[11]  Xing Gao,et al.  Resistance distances and the Kirchhoff index in Cayley graphs , 2011, Discret. Appl. Math..

[12]  Abhishek Dhar,et al.  Distribution of sizes of erased loops for loop-erased random walks , 1997 .

[13]  Jinde Cao,et al.  Cascade of failures in interdependent networks with different average degree , 2014 .

[14]  János Kertész,et al.  Geometry of minimum spanning trees on scale-free networks , 2003 .

[15]  Prabhakar Raghavan,et al.  The electrical resistance of a graph captures its commute and cover times , 2005, computational complexity.

[16]  István Lukovits,et al.  Resistance distance in regular graphs , 1999 .

[17]  Zuhe Zhang,et al.  Some physical and chemical indices of clique-inserted lattices , 2013, 1302.5932.

[18]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[19]  Stephan G. Wagner,et al.  Resistance Scaling and the Number of Spanning Trees in Self-Similar Lattices , 2011 .

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

[21]  Jia-Bao Liu,et al.  Asymptotic Laplacian-energy-like invariant of lattices , 2014, Appl. Math. Comput..

[22]  Jinde Cao,et al.  A note on ‘some physical and chemical indices of clique-inserted lattices’ , 2014 .

[23]  A. Kami'nska,et al.  Mean first passage time for a Markovian jumping process , 2007 .

[24]  Shuigeng Zhou,et al.  Enumeration of spanning trees in a pseudofractal scale-free web , 2010, 1008.0267.

[25]  Jurgen Kurths,et al.  Synchronization in complex networks , 2008, 0805.2976.

[26]  Shuigeng Zhou,et al.  Explicit determination of mean first-passage time for random walks on deterministic uniform recursive trees. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  Baoyu Hou,et al.  Applications of Laplacian spectra for extended Koch networks , 2012 .

[28]  Zhongzhi Zhang,et al.  Spanning trees in a fractal scale-free lattice. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  Shilpa Chakravartula,et al.  Complex Networks: Structure and Dynamics , 2014 .

[30]  Guanrong Chen,et al.  Mean first-passage time for random walks on undirected networks , 2011, 1111.1500.