no. 5
A bound for the A α -spectral radius of a connected graph after vertex deletion
RAIRO. Operations Research, Tome 56 (2022) no. 5, pp. 3635-3642

G is a simple connected graph with adjacency matrix A(G) and degree diagonal matrix D(G). The signless Laplacian matrix of G is defined as Q(G) = D(G) + A(G). In 2017, Nikiforov [1] defined the matrix A α ( G ) = α D ( G ) + ( 1 - α ) A ( G ) ) =  − for  ∈ [0,1]. The A α -spectral radius of G is the maximum eigenvalue of $$ (G). In 2019, Liu et al. [2] defined the matrix $$(G) as $$ (G) = kD(G) + A(G), for k ∈ ℝ. In this paper, we present a new type of lower bound for the $$-spectral radius of a graph after vertex deletion. Furthermore, we deduce some corollaries on $$ (G), A(G), Q(G) matrices.

DOI : 10.1051/ro/2022176
Classification : 05C50, 15A18
Keywords: Spectral radius, eigenvalue, eigenvector, adjacency matrix 
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     title = {A bound for the $A_{\alpha}$-spectral radius of a connected graph after vertex deletion},
     journal = {RAIRO. Operations Research},
     pages = {3635--3642},
     year = {2022},
     publisher = {EDP-Sciences},
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Wang, Chunxiang; She, Tao. A bound for the $A_{\alpha}$-spectral radius of a connected graph after vertex deletion. RAIRO. Operations Research, Tome 56 (2022) no. 5, pp. 3635-3642. doi: 10.1051/ro/2022176

[1] V. Nikiforov, Merging the 𝐀 -and 𝐐 -spectral theories. Appl. Anal. Discrete Math. 11 (2017) 81–107. | MR | Zbl | DOI

[2] M. Liu, H.-J. Lai and K. Ch. Das, Spectral results on hamiltonian problem. Discrete Math. 342 (2019) 1718–1730. | MR | Zbl | DOI

[3] D. B. West, Introduction to Graph Theory. 2nd edn. (2001). | MR

[4] V. Nikiforov, G. Pastén, O. Rojo and R. L. Soto, On the A α -spectra of trees. Linear Algebra Appl. 520 (2017) 286–305. | MR | Zbl | DOI

[5] V. Nikiforov and O. Rojo, A note on the positive semidefinitness of A α ( G ) . Linear Algebra Appl. 519 (2017) 156–163. | MR | Zbl | DOI

[6] V. Nikiforov and O. Rojo, On the α -index of graphs with pendent paths. Linear Algebra Appl. 550 (2018) 87–104. | MR | Zbl | DOI

[7] H. Lin, X. Huang and J. Xue, A Note on the A α -spectral Radius of Graphs (2018). | MR | Zbl

[8] D. Li, Y. Chen and J. Meng, The A α -spectral radius of trees and unicyclic graphs with given degree sequence. Appl. Math. Comput. 363 (2019). | MR | Zbl

[9] S. Wang, D. Wong and F. Tian, Bounds for the largest and the smallest A α eigenvalues of a graph in terms of vertex degrees. Linear Algebra Appl. 590 (2020) 210–223. | MR | Zbl | DOI

[10] J. M. Guo, Z. W. Wang and X. Li, Sharp upper bounds of the spectral radius of a graph. Discrete Math. 342 (2019) 2559–2563. | Zbl | MR | DOI

[11] S. Sun and K. C. Das, A conjecture on the spectral radius of graphs. Linear Algebra Appl. (2019) 588. | MR | Zbl

[12] M. Zhai, H. Lin and J. Shu, Spectral Extrema of Graphs with Fixed Size: Cycles and Complete Bipartite Graphs (2021). | MR | Zbl

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