@article{Fakieh2024, 
author = {Wafaa Fakieh and Zakeiah Alkhamisi and Hanaa Alashwali},
title = {On the        A                  α        −            -spectra of graphs and the relation between        A          α      - and        A                  α        −            -spectra},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {2},
pages = {4587-4603},
keywords = {Laplacian, singnless Laplacian, Aα-spectral radius, Aα-matrix, sum of powers of Aα-eigenvalues},
url = {https://www.sciopen.com/article/10.3934/math.2024221},
doi = {10.3934/math.2024221},
abstract = {Let    G be a graph with adjacency matrix    A  (  G  ), and let    D  (  G  ) be the diagonal matrix of the degrees of    G. For any real number    α  ∈  [  0  ,  1  ], Nikiforov defined the        A          α      -matrix of    G as         A          α        (  G  )  =  α  D  (  G  )  +  (  1  −  α  )  A  (  G  )  .The eigenvalues of the matrix        A          α        (  G  ) form the        A          α      -spectrum of    G. The        A          α      -spectral radius of    G is the largest eigenvalue of        A          α        (  G  ) denoted by        ρ    α    (  G  ). In this paper, we propose the        A                  α        −            -matrix of    G as         A                  α        −              (  G  )  =  α  D  (  G  )  +  (  α  −  1  )  A  (  G  )  ,        0  ≤  α  ≤  1.Let the        A                  α        −            -spectral radius of    G be denoted by        λ                  α        −              (  G  ), and let        S          β              α        (  G  ) and        S          β                      α        −              (  G  ) be the sum of the        β          t      h       powers of the        A          α       and        A                  α        −             eigenvalues of    G, respectively. We determine the        A                  α        −            -spectra of some graphs and obtain some bounds of the        A                  α        −            -spectral radius. Moreover, we establish a relationship between the        A          α      -spectral radius and        A                  α        −            -spectral radius. Indeed, for    α  ∈  (      1    2    ,  1  ), we show that        λ                  α        −              ≤      ρ    α  , and we prove that if    G is connected, then the equality holds if and only if    G is bipartite. Employing this relation, we obtain some upper bounds of        λ                  α        −              (  G  ), and we prove that the        A                  α        −            -spectrum and        A    α  -spectrum are equal if and only if    G is a bipartite connected graph. Furthermore, we generalize the relation established by S. Akbari et al. in    (  2010  ) as follows: for    α  ∈  [      1    2    ,  1  ), if          0  <  β  ≤  1 or      2  ≤  β  ≤  3, then        S          β              α        (  G  )  ≥      S          β                      α        −              (  G  )  , and if      1  ≤  β  ≤  2, then        S          β              α        (  G  )  ≤      S          β                      α        −              (  G  )  , where the equality holds if and only if    G is a bipartite graph such that    β  ∉  {  1  ,  2  ,  3  }  .}
}