@article{Hu2022, 
author = {Meiling Hu and Shuli Li},
title = {Cospectral graphs for the normalized Laplacian},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {3},
pages = {4061-4067},
keywords = {normalized Laplacian, cospectral, tridiagonal matrix, normalized Laplacian characteristic polynomial},
url = {https://www.sciopen.com/article/10.3934/math.2022224},
doi = {10.3934/math.2022224},
abstract = {Let    G  (      a    1    ,      a    2    ,  …  ,      a    k    ) be a simple graph with vertex set    V  (  G  )  =      V    1    ∪      V    2    ∪  ⋯  ∪      V    k   and edge set    E  (  G  )  =  {  (  u  ,  v  )      |    u  ∈      V    i    ,  v  ∈      V          i      +      1        ,  i  =  1  ,  2  ,  …  ,  k  −  1  }, where        |        V    i        |    =      a    i    >  0 for    1  ≤  i  ≤  k and        V    i    ∩      V    j    =  ∅ for    i  ≠  j. Given two positive integers    k and    n, and    k  −  2 positive rational numbers        t    2    ,      t    3    ,  …  ,      t          ⌈      k              /            2      ⌉       and        t    2    ′    ,      t    3    ′    ,  …  ,      t          ⌊      k              /            2      ⌋        ′  , let    Υ  (  n  ;  k      )    t                  t        ′              =  {  G  (      a    1    ,      a    2    ,  …  ,      a    k    )      |        ∑          i      =      1        k        a    i    =  n  ,      a          2      i      −      1        =      t          i            a    1    ,      a          2      j        =      t    j    ′        a    2    ,  i  =  2  ,  3  ,  …  ,  ⌈  k      /    2  ⌉  ,    j  =  2  ,  3  ,  …  ,  ⌊  k      /    2  ⌋  ;  t  =  (      t    2    ,      t    3    ,  …  ,      t          ⌈      k              /            2      ⌉        )  ,      t    ′    =  (      t    2    ′    ,      t    3    ′    ,  …  ,      t          ⌊      k              /            2      ⌋        ′    )  ;      a    s    ∈  N  ,  1  ≤  s  ≤  k  }, where    N is the set of positive integers. In this paper, we prove that all graphs in    Υ  (  n  ;  k      )    t                  t        ′             are cospectral with respect to the normalized Laplacian if it is not an empty set.}
}