title: Coherence in a family of tree networks with an application of Laplacian spectrum.
abstract: In this paper, we study consensus problems in a family of tree networks and investigate first and second order consensus denoted as network coherence characterized by Laplacian spectrum. According to the tree structures, we obtain the recursive relationships of Laplacian matrix and its Laplacian eigenvalues at two successive generations. We then obtain the analytical expressions for the sum of the reciprocals and the square reciprocals of all nonzero Laplacian eigenvalues. Finally, we calculate first and second order network coherence and see that the scalings of first and second order coherence with network size N are lnN and N, which are smaller than some studied tree graphs, such as Peano basin fractal, T-graph, and generalized Vicsek fractal.
authors.affiliation: Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China.
authors.forename: Weigang
authors.initials: W
authors.lastname: Sun
authors.affiliation: Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China.
authors.forename: Qingyan
authors.initials: Q
authors.lastname: Ding
authors.affiliation: Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China.
authors.forename: Jingyuan
authors.initials: J
authors.lastname: Zhang
authors.affiliation: Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China.
authors.forename: Fangyue
authors.initials: F
authors.lastname: Chen