The A α -spectra of graphs Huiqiu Lin Department of Mathematics East China University of Science and Technology Joint work with Xiaogang Liu, Jinlong Shu and Jie Xue 2019-04-28
Outline 1 Basic Notations Some known results 2 Our results 3 H. Lin The A α -spectra of graphs 2019-04-28 1 / 25
Basic Notations Outline 1 Basic Notations Some known results 2 Our results 3 H. Lin The A α -spectra of graphs 2019-04-28 2 / 25
Basic Notations Basic Notations • Let G be a graph with vertex set { v 1 , v 2 , . . . , v n } . The degree of the vertex v i is denoted by d i . • Adjacency matrix: A ( G ) = ( a ij ) n × n , � 1 if v i ∼ v j , a ij = 0 if v i ≁ v j . • Degree matrix: D ( G ) = diag ( d 1 , d 2 , . . . , d n ) • Laplacian matrix: L ( G ) = D ( G ) − A ( G ) • Signless Laplacian matrix: Q ( G ) = D ( G ) + A ( G ) - Laplacian matrix and signless Laplacian matrix are all positive semi-definite, they contain the same eigenvalues if G is a bipartite graph. - The Laplacian spectrum and signless Laplacian spectrum are given by the adjacency spectrum if G is a regular graph. H. Lin The A α -spectra of graphs 2019-04-28 3 / 25
Basic Notations • In extremal spectral graph theory, there are many similar conclusions with respect to A -matrix and Q -matrix. Graph type Objective Extremal graph unicycle graphs maximize the spectral radius same / signless Laplaican spectral radius bicyclic graphs maximize the spectral radius same / signless Laplaican spectral radius graphs with maximize the spectral radius same given diameter /signless Laplaican spectral radius graphs with minimize the spectral radius same given clique number /signless Laplaican spectral radius ... ... ... • However, there are also a lot of differences between adjacency spectra and signless Laplacian spectra, and the research on Q ( G ) has shown that it is a remarkable matrix, unique in many respects. H. Lin The A α -spectra of graphs 2019-04-28 4 / 25
Basic Notations In order to study both similarities and differences between A ( G ) and Q ( G ), Nikiforov [1] introduced a new matrix A α ( G ): For a real number α ∈ [0 , 1], the A α -matrix of G is A α ( G ) = α D ( G ) + (1 − α ) A ( G ) , where A ( G ) is the adjacency matrix and D ( G ) is the degree diagonal matrix of G . • A α -eigenvalues: λ 1 ( A α ( G )) ≥ λ 2 ( A α ( G )) ≥ · · · ≥ λ n ( A α ( G )) • A α -spectral radius: λ 1 ( A α ( G )) - if α = 0, then A α ( G ) = A ( G ) - if α = 1 / 2, then A α ( G ) = 1 2 Q ( G ) - if α = 1 then A α ( G ) = D ( G ) [1] V. Nikiforov, Merging the A- and Q-spectral theories , Appl. Anal. Discrete Math. 11 (2017) 81-107. H. Lin The A α -spectra of graphs 2019-04-28 5 / 25
Some known results Outline 1 Basic Notations Some known results 2 Our results 3 H. Lin The A α -spectra of graphs 2019-04-28 6 / 25
Some known results For a graph G , the A α -eigenvalues are increasing in α . Theorem (Nikiforov 2017) Let 1 ≥ α ≥ β ≥ 0 . If G is a graph of order n, then λ k ( A α ( G )) ≥ λ k ( A β ( G )) for any k ∈ [ n ] . If G is connected, then inequality is strict, unless k = 1 and G is regular. Take α = 0 , 1 2 , 1, thus ( A 0 ( G ) = D ( G ) , A 1 ( G ) = A ( G )) λ k ( D ( G )) ≥ λ k ( A 1 2 ( G )) ≥ λ k ( A ( G )) . Take k = 1, thus 2∆( G ) ≥ q ( G ) ≥ 2 ρ ( G ) , where ∆( G ) is the maximum degree, q ( G ) is the signless Laplacian spectral radius and ρ ( G ) is the spectral radius of G , respectively. H. Lin The A α -spectra of graphs 2019-04-28 7 / 25
Some known results The positive semidefiniteness of A α ( G ) Note that the signless Laplacian matrix is positive semidefinite, that is, A 1 2 ( G ) is positive semidefinite. Theorem ([2], Nikiforov and Rojo 2017) Let G be a graph. If α ≥ 1 / 2 , then A α ( G ) is positive semidefinite. If α > 1 / 2 and G has no isolated vertices, then A α ( G ) is positive definite. It is natural to consider the following problem. Problem For a graph G, determine the minimum α such that A α ( G ) is positive semidefinite. [2] V. Nikiforov, O. Rojo, A note on the positive semidefiniteness of A α ( G ), LAA 519 (2017) 156-163. H. Lin The A α -spectra of graphs 2019-04-28 8 / 25
Some known results Let α 0 ( G ) be the smallest α for which A α ( G ) is positive semidefinite. Nikiforov and Rojo [2] showed that: - α 0 ( G ) ≤ 1 / 2; - if G is k -regular then − λ min ( A ( G )) α 0 ( G ) = k − λ min ( A ( G )) where λ min ( A ( G )) is the smallest eigenvalue of A ( G ); - G contains a bipartite component if and only if α 0 ( G ) = 1 / 2; - if G is r -colorable, then α 0 ( G ) ≥ 1 / r . H. Lin The A α -spectra of graphs 2019-04-28 9 / 25
Some known results Theorem (Nikiforov 2017) Let r ≥ 2 and G be an r-chromatic graph of order n. (1) If α < 1 − 1 / r, then λ 1 ( A α ( G )) ≤ λ 1 ( A α ( T r ( n ))) , with equality if and only if G ∼ = T r ( n ) (r -partite Tur´ an graph ) (2) If α > 1 − 1 / r, then λ 1 ( A α ( G )) ≤ λ 1 ( A α ( S n , r − 1 )) , with equality if and only if G ∼ = S n , r − 1 ( K r − 1 ∨ K c n − r + 1 ). (3) If α = 1 − 1 / r, then λ 1 ( A α ( G )) ≤ (1 − 1 / r ) n , with equality if and only if G is a complete r-partite graph. H. Lin The A α -spectra of graphs 2019-04-28 10 / 25
Some known results Theorem (Nikiforov 2017) Let r ≥ 2 and G be a K r +1 -free graph of order n. (1) If α < 1 − 1 / r, then λ 1 ( A α ( G )) ≤ λ 1 ( A α ( T r ( n ))) , with equality if and only if G ∼ = T r ( n ) . (2) If α > 1 − 1 / r, then λ 1 ( A α ( G )) ≤ λ 1 ( A α ( S n , r − 1 )) , with equality if and only if G ∼ = S n , r − 1 . (3) If α = 1 − 1 / r, then λ 1 ( A α ( G )) ≤ (1 − 1 / r ) n , with equality if and only if G is a complete r-partite graph. The techniques used here are partially from [ He, Jin and Zhang: Sharp bounds for the signless Laplacian spectral radius in terms of clique number, LAA 438 (2013) 3851-3861. ] H. Lin The A α -spectra of graphs 2019-04-28 11 / 25
Some known results Theorem ([3], Nikiforov, Past´ en, Rojo and Soto 2017) If T is a tree of order n, then λ 1 ( A α ( T )) ≤ λ 1 ( A α ( K 1 , n − 1 )) . Equality holds if and only if T ∼ = K 1 , n − 1 . Theorem ([3], Nikiforov, Past´ en, Rojo and Soto 2017) If G is a connected graph of order n, then λ 1 ( A α ( G )) ≥ λ 1 ( A α ( P n )) . Equality holds if and only if G ∼ = P n . [3] V. Nikiforov, G. Past´ en, O. Rojo, R.L. Soto, On the A α ( G ) -spectra of trees , LAA 520 (2017) 286-305. H. Lin The A α -spectra of graphs 2019-04-28 12 / 25
Our results Outline 1 Basic Notations Some known results 2 Our results 3 H. Lin The A α -spectra of graphs 2019-04-28 13 / 25
Our results Let G be a connected graph and u , v be two distinct vertices of V ( G ). Let G p , q ( u , v ) be the graph obtained by attaching the paths P p to u and P q to v . The following problem is inspired by the results of Li and Feng [5]. Problem ([4] Nikiforov, Rojo 2018) For which connected graphs G the following statement is true: Let α ∈ [0 , 1) and let u and v be non-adjacent vertices of G of degree at least 2. If q ≥ 1 and p ≥ q + 2, then ρ α ( G p , q ( u , v )) < ρ α ( G p − 1 , q +1 ( u , v )). [4] V. Nikiforov, O. Rojo, On the α -index of graphs with pendent paths. Linear Algebra Appl. 550 (2018) 87-104. [5] Q. Li, K. Feng, On the largest eigenvalue of graphs, Acta Math. Appl. Sin. 2 (1979) 167-175. H. Lin The A α -spectra of graphs 2019-04-28 14 / 25
Our results Let G be a connected graph and u , v ∈ V ( G ) with d ( u ) , d ( v ) ≥ 2. Suppose that u and v is connected by a path w 0 (= v ) w 1 · · · w s − 1 w s (= u ) where d ( w i ) = 2 for 1 ≤ i ≤ s − 1. Let G p , s , q ( u , v ) be the graph obtained by attaching the paths P p to u and P q to v . Theorem (Lin, Huang, Xue 2018) Let 0 ≤ α < 1. If p − q ≥ max { s + 1 , 2 } , then ρ α ( G p − 1 , s , q +1 ( u , v )) > ρ α ( G p , s , q ( u , v )). H. Lin, X. Huang, J. Xue, A note on the A α -spectral radius of graphs, Linear Algebra Appl. 557 (2018) 430–437. H. Lin The A α -spectra of graphs 2019-04-28 15 / 25
Our results The above theorem implies that the following conjecture is true. Conjecture (Nikiforov, Rojo 2018) Let 0 ≤ α < 1 and s = 0 , 1. If p ≥ q + 2, then ρ α ( G p , s , q ( u , v )) < ρ α ( G p − 1 , s , q +1 ( u , v )). It needs to be noticed that, the above conjecture is independently confirmed by Guo and Zhou [6]. [6] H. Guo, B. Zhou, On the α -spectral radius of graphs, arXiv:1805.03456. H. Lin The A α -spectra of graphs 2019-04-28 16 / 25
Our results The k -th largest eigenvalue Theorem (Lin, Xue, Shu 2018) Let G be a graph with n vertices. If α ≥ 1 / 2 and e / ∈ E ( G ), then λ k ( A α ( G + e )) ≥ λ k ( A α ( G )) . ⊲ Using this theorem, we get an upper bound on the A α -eigenvalue when α ≥ 1 / 2: λ k ( A α ( G )) ≤ λ k ( A α ( K n )) = α n − 1 . Problem Which graphs satisfy λ k ( A α ( G )) = α n − 1 ? H. Lin, J. Xue, J. Shu, On the A α -spectra of graphs, Linear Algebra Appl. 556 (2018) 210–219. H. Lin The A α -spectra of graphs 2019-04-28 17 / 25
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