A BOUND FOR THE 𝐴 𝛼 -SPECTRAL RADIUS OF A CONNECTED GRAPH AFTER VERTEX DELETION

. 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 ) = 𝛼𝐷 ( 𝐺 ) + (1 − 𝛼 ) 𝐴 ( 𝐺 ) for 𝛼 ∈ [0 , 1] . The 𝐴 𝛼 -spectral radius of 𝐺 is the maximum eigenvalue of 𝐴 𝛼 ( 𝐺 ). In 2019, Liu et al. [2] defined the matrix Θ 𝑘 ( 𝐺 ) as Θ 𝑘 ( 𝐺 ) = 𝑘𝐷 ( 𝐺 ) + 𝐴 ( 𝐺 ), for 𝑘 ∈ R . In this paper, we present a new type of lower bound for the 𝐴 𝛼 -spectral radius of a graph after vertex deletion. Furthermore, we deduce some corollaries on Θ 𝑘 ( 𝐺 ) , 𝐴 ( 𝐺 ) , 𝑄 ( 𝐺 ) matrices.


Introduction and preliminaries
We consider non-empty simple connected graph  with vertex set  () and edge set () throughout this paper.Let  () = { 1 ,  2 , . . .,   }.If any pair of vertices   and   are adjacent, then we write     ∈ () or   ∼   .For a vertex   ∈  (), the neighborhood of   is the set  (  ) =   (  ) = { ∈  () :  ∼   }, and   (  ) denotes the degree of   with   (  ) = | (  )|.Let   =   (  ) if there is no ambiguity.Let   be any fixed subset of  () containing  vertices.For   ⊆  () with |  | = , let [  ] be the subgraph of  induced by   ,  −   be the subgraph induced by  () −   .Let  ∨  denote the graph obtained from the disjoint union  +  by adding all edges between graph  and graph .A regular graph with vertices of degree  is called a -regular graph.Let   ,  , denote the clique and complete bipartite graph respectively and  1,−1 be the star of order .
For other undefined notations and terminologies, refer to [3].Many scholars already succeeded in finding bounds for the   -spectral radius.For more results in this direction, readers can refer to a survey [1] by Nikiforov and some other articles [4][5][6][7][8][9].In 2019, Guo et al. [10] and Sun et al. [11] presented a relation between  1 () and  1 ( −   ) for adjacency matrices () and ( −   ), where   is a vertex of .The better bound given by [11] is shown as follow Since ( 1) is well used in analyzing the graph structure (see [12]), we try to extend the above inequation to more matrices, such as   (), Θ  (), ().Using some different methods from [10,11], we get the results in this paper.As far as the authors know, this topic has not been explored elsewhere.

Main results
Theorem 2.1.Let  ∈ [0, 1) and let   be a vertex of a connected graph  with degree   .Then with equality holding if and only if  is the join of the vertex   and a regular graph of order  − 1.
Theorem 2.1 will be proved in Section 3.
Corollary 2.2.Let  ∈ [0, +∞) and let  be a vertex of a connected graph  with degree ().Then with equality holding if and only if  is the join of the vertex  and a regular graph of order  − 1.
By setting  = 0 in (2) we obtain the following corollary.
Corollary 2.3.Let   be a vertex of a connected graph  with degree   .Then with equality holding if and only if  is the join of the vertex   and a regular graph of order  − 1.
When   > By setting  = 1 2 in (2) we obtain the following corollary.
Corollary 2.4.Let   be a vertex of a connected graph  with degree   , and let () be the signless Laplacian matrix of .Then with equality holding if and only if  is the join of the vertex   and a regular graph of order  − 1.
Define Φ(, , ) Corollary 2.5.Let  be the join of the vertex   and a -regular graph of order  − 1,  ∈ [0, 1).Then Proof.() Since  −   is a -regular graph, it is obvious that   ( −   ) = .By Theorem 2.1, we have We can also calculate   by equitable quotient matrices for the graphs in Corollary 2.5.

Proof of
with equality holding if and only if   is an empty graph.
Proof.Since   = () + (1 − )(), we have Suppose  ′  is the ()th row ()th column element of   (  ), then Let a  be the column vector ( 1 ,  2 , . . .,   )  and e  be the th basis column vector (0, . . ., 0, 1, 0, . . ., 0)  , where only the th component is 1.We have Hence Note that x is a positive unit eigenvector, we can see that the equality holds if and only if   is an empty graph.
Lemma 3.3.Let  ∈ [0, 1), let x be the positive eigenvector of   () corresponding to   with x  x = 1, let   be the th component of x, and let   be the th vertex of .Then Proof.Let   =  () −   , then   is an empty graph.By Lemma 3.2, we have Lemma 3.4.Let  ∈ [0, 1), let x be the positive eigenvector of   () corresponding to   with x  x = 1, let   be the th component of x, and let   be the th vertex of .Then with equality holding if and only if  is the join of the vertex   and a regular graph of order  − 1. Proof.Since we have That is By Lemma 3.4 we get is a increasing function on   ∈ (0, 1),   −   > 0 if  ∈ [0, 1) (by Lem.3.1), combining (11)with (12), we have This completes the proof of inequality (2).On one hand, the equality in (2) holds implying the equality in ( 12) holds.By Lemma 3.4,  is the join of the vertex   and a regular graph of order  − 1.
On the other hand, if  is the join of the vertex   and a regular graph of order  − 1, then the equality in (7) holds, subsequently the equalities in ( 6) and ( 11) hold.Combining with Lemma 3.4, we get that the equalities in ( 8) and ( 12) hold, and the equality in (2) holds.
This journal is currently published in open access under a Subscribe-to-Open model (S2O).S2O is a transformative model that aims to move subscription journals to open access.Open access is the free, immediate, online availability of research articles combined with the rights to use these articles fully in the digital environment.We are thankful to our subscribers and sponsors for making it possible to publish this journal in open access, free of charge for authors.

x
(  −   (  ))x (10) Equality in(10)holds if and only if  (  ) =  () −   . Othis basis, the equality in (9) holds if and only if all the   is same where  ∈ {1, 2, . .., } and  ̸ = , if and only if  −   is a regular graph.Proof of Theorem 2.1.Let x be the positive eigenvector of   () corresponding to   with x  x = 1, and let   be the th component of x.By Lemma 3.3, we obtain