Graph spectral conditions and structural properties Hong-Jian Lai - PowerPoint PPT Presentation

Cioaba’s Problem Let G be a d -regular graph. Theorem (Cioaba and Wong, LAA 2012) Assume that 3 4 ≤ d . If λ 2 ( G ) < d − d +1 , then τ ( G ) ≥ 2 . Theorem (Cioaba and Wong, LAA 2012) Assume that 5 6 ≤ d . If λ 2 ( G ) < d − d +1 , then τ ( G ) ≥ 3 . – p. 11/39

Cioaba’s Problem Let G be a d -regular graph. Theorem (Cioaba and Wong, LAA 2012) Assume that 3 4 ≤ d . If λ 2 ( G ) < d − d +1 , then τ ( G ) ≥ 2 . Theorem (Cioaba and Wong, LAA 2012) Assume that 5 6 ≤ d . If λ 2 ( G ) < d − d +1 , then τ ( G ) ≥ 3 . Conjecture (Cioaba and Wong, LAA 2012) Assume that 2 ≤ 2 k ≤ d . If λ 2 ( G ) < d − 2 k − 1 d +1 , then τ ( G ) ≥ k . – p. 11/39

Improvements in JGT, 2016 Can we work on generic graphs in stead of regular graphs? – p. 12/39

Improvements in JGT, 2016 Can we work on generic graphs in stead of regular graphs? Let G be graph with δ ( G ) = δ and k > 0 be an integer. – p. 12/39

Improvements in JGT, 2016 Can we work on generic graphs in stead of regular graphs? Let G be graph with δ ( G ) = δ and k > 0 be an integer. . Li, S. Yao and HJL, JGT 2016) If δ ≥ 4 Theorem (X. Gu, P 3 and λ 2 ( G ) < δ − δ +1 , then τ ( G ) ≥ 2 . – p. 12/39

Improvements in JGT, 2016 Can we work on generic graphs in stead of regular graphs? Let G be graph with δ ( G ) = δ and k > 0 be an integer. . Li, S. Yao and HJL, JGT 2016) If δ ≥ 4 Theorem (X. Gu, P 3 and λ 2 ( G ) < δ − δ +1 , then τ ( G ) ≥ 2 . Theorem (X. Gu, P . Li, S. Yao and HJL, JGT 2016) If δ ≥ 6 5 and λ 2 ( G ) < δ − δ +1 , then τ ( G ) ≥ 3 . – p. 12/39

Improvements in JGT, 2016 Let G be graph with δ ( G ) = δ and k > 0 be an integer. – p. 13/39

Improvements in JGT, 2016 Let G be graph with δ ( G ) = δ and k > 0 be an integer. Theorem (Cioaba, LAA 2010) If G is d -regular, d ≥ 2 k , and λ 2 ( G ) < d − 4 k − 2 d +1 , then τ ( G ) ≥ k . – p. 13/39

Improvements in JGT, 2016 Let G be graph with δ ( G ) = δ and k > 0 be an integer. Theorem (Cioaba, LAA 2010) If G is d -regular, d ≥ 2 k , and λ 2 ( G ) < d − 4 k − 2 d +1 , then τ ( G ) ≥ k . . Li, S. Yao and HJL, JGT 2016) If δ ≥ 2 k Theorem (X. Gu, P and λ 2 ( G ) < δ − 3 k − 1 δ +1 , then τ ( G ) ≥ k . – p. 13/39

Improvements in JGT, 2016 Let G be graph with δ ( G ) = δ and k > 0 be an integer. Theorem (Cioaba, LAA 2010) If G is d -regular, d ≥ 2 k , and λ 2 ( G ) < d − 4 k − 2 d +1 , then τ ( G ) ≥ k . . Li, S. Yao and HJL, JGT 2016) If δ ≥ 2 k Theorem (X. Gu, P and λ 2 ( G ) < δ − 3 k − 1 δ +1 , then τ ( G ) ≥ k . Conjecture Let G be graph with δ ( G ) = δ , and 4 ≤ 2 k ≤ δ . If λ 2 ( G ) < δ − 2 k − 1 δ +1 , then τ ( G ) ≥ k . – p. 13/39

Over view of progresses Conjecture ( k, δ ) Let G be graph with δ ( G ) = δ and 2 k ≤ δ . If λ 2 ( G ) < δ − 2 k − 1 δ +1 , then τ ( G ) ≥ k . – p. 14/39

Over view of progresses Conjecture ( k, δ ) Let G be graph with δ ( G ) = δ and 2 k ≤ δ . If λ 2 ( G ) < δ − 2 k − 1 δ +1 , then τ ( G ) ≥ k . Let G be graph on n vertices with δ = δ ( G ) ≥ 2 k ≥ 4 . – p. 14/39

Over view of progresses Conjecture ( k, δ ) Let G be graph with δ ( G ) = δ and 2 k ≤ δ . If λ 2 ( G ) < δ − 2 k − 1 δ +1 , then τ ( G ) ≥ k . Let G be graph on n vertices with δ = δ ( G ) ≥ 2 k ≥ 4 . Theorem (G. Li and L. Shi, LAA 2013; Y. Hong, Q. Liu, and HJL, LAA 2014) For any integer k ≥ 2 and δ ≥ 2 k , there exists an integer N = N ( k, δ ) such that if n ≥ N , then Conjecture( k, δ ) holds, – p. 14/39

Over view of progresses Conjecture (Gu et al.) Let G be a graph with minimum degree δ ≥ 2 k ≥ 4 . If λ 2 ( G ) < δ − 2 k − 1 δ +1 , then τ ( G ) ≥ k . – p. 15/39

Over view of progresses Conjecture (Gu et al.) Let G be a graph with minimum degree δ ≥ 2 k ≥ 4 . If λ 2 ( G ) < δ − 2 k − 1 δ +1 , then τ ( G ) ≥ k . It is a theorem. (Y. Hong, Q. Liu, Gu, and HJL, LAA 2014) – p. 15/39

Over view of progresses Conjecture (Gu et al.) Let G be a graph with minimum degree δ ≥ 2 k ≥ 4 . If λ 2 ( G ) < δ − 2 k − 1 δ +1 , then τ ( G ) ≥ k . It is a theorem. (Y. Hong, Q. Liu, Gu, and HJL, LAA 2014) How about Laplacian eigenvalues? (Algebraic connectivity)? – p. 15/39

Over view of progresses Conjecture (Gu et al.) Let G be a graph with minimum degree δ ≥ 2 k ≥ 4 . If λ 2 ( G ) < δ − 2 k − 1 δ +1 , then τ ( G ) ≥ k . It is a theorem. (Y. Hong, Q. Liu, Gu, and HJL, LAA 2014) How about Laplacian eigenvalues? (Algebraic connectivity)? How about signless Laplacian eigenvalues? – p. 15/39

Over view of progresses A = A ( G ) : = adjacency matrix of G . – p. 16/39

Over view of progresses A = A ( G ) : = adjacency matrix of G . D = D ( G ) : = degree diagonal matrix of G . – p. 16/39

Over view of progresses A = A ( G ) : = adjacency matrix of G . D = D ( G ) : = degree diagonal matrix of G . A − D gives Laplacian eigenvalues. – p. 16/39

Over view of progresses A = A ( G ) : = adjacency matrix of G . D = D ( G ) : = degree diagonal matrix of G . A − D gives Laplacian eigenvalues. D + A gives signless Laplacian eigenvalues. – p. 16/39

Over view of progresses A = A ( G ) : = adjacency matrix of G . D = D ( G ) : = degree diagonal matrix of G . A − D gives Laplacian eigenvalues. D + A gives signless Laplacian eigenvalues. a : = a real number. – p. 16/39

Over view of progresses A = A ( G ) : = adjacency matrix of G . D = D ( G ) : = degree diagonal matrix of G . A − D gives Laplacian eigenvalues. D + A gives signless Laplacian eigenvalues. a : = a real number. λ 1 ( G, a ) ≥ λ 2 ( G, a ) ≥ · · · ≥ λ n ( G, a ) are eigenvalues of aD + A . – p. 16/39

Over view of progresses λ 1 ( G, a ) ≥ λ 2 ( G, a ) ≥ · · · ≥ λ n ( G, a ) are eigenvalues of aD + A . – p. 17/39

Over view of progresses λ 1 ( G, a ) ≥ λ 2 ( G, a ) ≥ · · · ≥ λ n ( G, a ) are eigenvalues of aD + A . Theorem. (Liu, Hong, Gu, HJL, LAA 2014) Let k be an integer and G be a graph of order n and minimum degree δ ≥ 2 k . If λ 2 ( G, a ) < ( a + 1) δ − 2 k − 1 δ +1 then τ ( G ) ≥ k . – p. 17/39

Over view of progresses λ 1 ( G, a ) ≥ λ 2 ( G, a ) ≥ · · · ≥ λ n ( G, a ) are eigenvalues of aD + A . Theorem. (Liu, Hong, Gu, HJL, LAA 2014) Let k be an integer and G be a graph of order n and minimum degree δ ≥ 2 k . If λ 2 ( G, a ) < ( a + 1) δ − 2 k − 1 δ +1 then τ ( G ) ≥ k . Choose different values of a ∈ { 0 , 1 , − 1 } . – p. 17/39

Over view of progresses λ i ( G ) : = the i th largest eigenvalue of A . µ i ( G ) : = the i th largest eigenvalue of D − A . q i ( G ) : = the i th largest eigenvalue of D + A . – p. 18/39

Over view of progresses λ i ( G ) : = the i th largest eigenvalue of A . µ i ( G ) : = the i th largest eigenvalue of D − A . q i ( G ) : = the i th largest eigenvalue of D + A . Theorem. (Liu, Hong, Gu, HJL, LAA 2014) (1) If λ 2 ( G ) < δ − 2 k − 1 δ +1 , then τ ( G ) ≥ k . (2) If q 2 ( G ) < 2 δ − 2 k − 1 δ +1 , then τ ( G ) ≥ k . (3) If µ n − 1 ( G ) > 2 k − 1 δ +1 , then τ ( G ) ≥ k . – p. 18/39

Outline of Proof of Cioaba-Wong Conjecture The U -Lemma. – p. 19/39

Outline of Proof of Cioaba-Wong Conjecture The U -Lemma. Quadratic Inequality. – p. 19/39

Outline of Proof of Cioaba-Wong Conjecture The U -Lemma. Quadratic Inequality. Proof of Cioaba-Wong Conjecture. – p. 19/39

Outline of Proof of Cioaba-Wong Conjecture U -Lemma Let G be a graph with minimum degree δ > 0 and ∅ � = U ⊂ V ( G ) . If d ( U ) ≤ δ − 1 , then | U | ≥ δ + 1 . – p. 20/39

Outline of Proof of Cioaba-Wong Conjecture U -Lemma Let G be a graph with minimum degree δ > 0 and ∅ � = U ⊂ V ( G ) . If d ( U ) ≤ δ − 1 , then | U | ≥ δ + 1 . Proof: d ( U ) ≤ δ − 1 means U has a vertex u ∈ U not incident with any edges in [ U, V − U ] . – p. 20/39

Outline of Proof of Cioaba-Wong Conjecture U -Lemma Let G be a graph with minimum degree δ > 0 and ∅ � = U ⊂ V ( G ) . If d ( U ) ≤ δ − 1 , then | U | ≥ δ + 1 . Proof: d ( U ) ≤ δ − 1 means U has a vertex u ∈ U not incident with any edges in [ U, V − U ] . N G ( u ) ⊆ U . – p. 20/39

Outline of Proof of Cioaba-Wong Conjecture U -Lemma Let G be a graph with minimum degree δ > 0 and ∅ � = U ⊂ V ( G ) . If d ( U ) ≤ δ − 1 , then | U | ≥ δ + 1 . Proof: d ( U ) ≤ δ − 1 means U has a vertex u ∈ U not incident with any edges in [ U, V − U ] . N G ( u ) ⊆ U . | U | ≥ |{ u } ∪ N G ( u ) | ≥ 1 + δ . – p. 20/39

Outline of Proof of Cioaba-Wong Conjecture Lemma (Quadratic Inequality) Let X, Y ⊂ V ( G ) with X ∩ Y = ∅ . If – p. 21/39

Outline of Proof of Cioaba-Wong Conjecture Lemma (Quadratic Inequality) Let X, Y ⊂ V ( G ) with X ∩ Y = ∅ . If λ 2 ( G, a ) ≤ ( a + 1) δ − max { d ( X ) | X | , d ( Y ) | Y | } , then – p. 21/39

Outline of Proof of Cioaba-Wong Conjecture Lemma (Quadratic Inequality) Let X, Y ⊂ V ( G ) with X ∩ Y = ∅ . If λ 2 ( G, a ) ≤ ( a + 1) δ − max { d ( X ) | X | , d ( Y ) | Y | } , then (( a + 1) δ − d ( X ) | [ X, Y ] | 2 ≥ | X | − λ 2 ( G, a )) · (( a + 1) δ − d ( Y ) | Y | − λ 2 ( G, a )) | X | · | Y | . – p. 21/39

Proof of Cioaba-Wong Conjecture (i) Theorem Let k be an integer and G be a graph of order n and minimum degree δ ≥ 2 k . If λ 2 ( G, a ) < ( a + 1) δ − 2 k − 1 δ +1 then τ ( G ) ≥ k . – p. 22/39

Proof of Cioaba-Wong Conjecture (i) Theorem Let k be an integer and G be a graph of order n and minimum degree δ ≥ 2 k . If λ 2 ( G, a ) < ( a + 1) δ − 2 k − 1 δ +1 then τ ( G ) ≥ k . Approach of the proof: For any partition ( V 1 , V 2 , . . . , V t ) , want to prove � 1 ≤ i<j ≤ t | [ V i , V j ] G | ≥ k ( t − 1) . – p. 22/39

Proof of Cioaba-Wong Conjecture (ii) Assume that d ( V 1 ) ≤ d ( V 2 ) ≤ . . . ≤ d ( V t ) . – p. 23/39

Proof of Cioaba-Wong Conjecture (ii) Assume that d ( V 1 ) ≤ d ( V 2 ) ≤ . . . ≤ d ( V t ) . If d ( V 1 ) ≥ 2 k , then � 1 ≤ i<j ≤ t | [ V i , V j ] G | ≥ kt . Assume d ( V 1 ) ≤ 2 k − 1 . – p. 23/39

Proof of Cioaba-Wong Conjecture (ii) Assume that d ( V 1 ) ≤ d ( V 2 ) ≤ . . . ≤ d ( V t ) . If d ( V 1 ) ≥ 2 k , then � 1 ≤ i<j ≤ t | [ V i , V j ] G | ≥ kt . Assume d ( V 1 ) ≤ 2 k − 1 . Let 1 ≤ s ≤ t be such that d ( V s ) ≤ 2 k − 1 and d ( V s +1 ) ≥ 2 k (if s < t ). – p. 23/39

Proof of Cioaba-Wong Conjecture (ii) Assume that d ( V 1 ) ≤ d ( V 2 ) ≤ . . . ≤ d ( V t ) . If d ( V 1 ) ≥ 2 k , then � 1 ≤ i<j ≤ t | [ V i , V j ] G | ≥ kt . Assume d ( V 1 ) ≤ 2 k − 1 . Let 1 ≤ s ≤ t be such that d ( V s ) ≤ 2 k − 1 and d ( V s +1 ) ≥ 2 k (if s < t ). By U-lemma, for 1 ≤ i ≤ s , | V i | ≥ δ + 1 . – p. 23/39

Proof of Cioaba-Wong Conjectur (iii) Assumption of Theorem, for 1 ≤ i ≤ s . λ 2 ( G, a ) < ( a + 1) δ − 2 k − 1 δ + 1 ≤ ( a + 1) δ − d ( V i ) | V i | . – p. 24/39

Proof of Cioaba-Wong Conjectur (iii) Assumption of Theorem, for 1 ≤ i ≤ s . λ 2 ( G, a ) < ( a + 1) δ − 2 k − 1 δ + 1 ≤ ( a + 1) δ − d ( V i ) | V i | . By Quadratic Inequality, for 2 ≤ i ≤ s , ( a + 1) δ − d ( V 1 ) | [ V 1 , V i ] | 2 � � ≥ | V 1 | − λ 2 ( G, a ) · ( a + 1) δ − d ( V i ) � � | V i | − λ 2 ( G, a ) | V 1 | · | V i | (2 k − 1 − d ( V 1 ))(2 k − 1 − d ( V i )) > (2 k − 1 − d ( V i )) 2 . ≥ – p. 24/39

Proof of Cioaba-Wong Conjectur (iii) Assumption of Theorem, for 1 ≤ i ≤ s . λ 2 ( G, a ) < ( a + 1) δ − 2 k − 1 δ + 1 ≤ ( a + 1) δ − d ( V i ) | V i | . By Quadratic Inequality, for 2 ≤ i ≤ s , ( a + 1) δ − d ( V 1 ) | [ V 1 , V i ] | 2 � � ≥ | V 1 | − λ 2 ( G, a ) · ( a + 1) δ − d ( V i ) � � | V i | − λ 2 ( G, a ) | V 1 | · | V i | (2 k − 1 − d ( V 1 ))(2 k − 1 − d ( V i )) > (2 k − 1 − d ( V i )) 2 . ≥ | [ V 1 , V i ] | > 2 k − 1 − d ( V i ) , for 2 ≤ i ≤ s . – p. 24/39

Proof of Cioaba-Wong Conjecture (iv) Thus | [ V 1 , V i ] | ≥ 2 k − d ( V i ) , for 2 ≤ i ≤ s . – p. 25/39

Proof of Cioaba-Wong Conjecture (iv) Thus | [ V 1 , V i ] | ≥ 2 k − d ( V i ) , for 2 ≤ i ≤ s . d ( V 1 ) ≥ � s i =2 | [ V 1 , V i ] | ≥ � s � � 2 k − d ( V i ) . i =2 – p. 25/39

Proof of Cioaba-Wong Conjecture (iv) Thus | [ V 1 , V i ] | ≥ 2 k − d ( V i ) , for 2 ≤ i ≤ s . d ( V 1 ) ≥ � s i =2 | [ V 1 , V i ] | ≥ � s � � 2 k − d ( V i ) . i =2 t s t � � � d ( V i ) = d ( V 1 ) + d ( V i ) + d ( V i ) i =1 i =2 i = s +1 ≥ 2 k ( s − 1) + 2 k ( t − s ) = 2 k ( t − 1) . – p. 25/39

References 1 A.E. Brouwer and W.H. Haemers, Spectra of Graphs, Springer Universitext 2012. (http://homepages.cwi.nl/ aeb/math/ipm.pdf). – p. 26/39

References 1 A.E. Brouwer and W.H. Haemers, Spectra of Graphs, Springer Universitext 2012. (http://homepages.cwi.nl/ aeb/math/ipm.pdf). 2 P . A. Catlin, H.-J. Lai and Y. Shao, Edge-connectivity and edge-disjoint spanning trees, Discrete Math., 309 (2009), 1033-1040. – p. 26/39

References 1 A.E. Brouwer and W.H. Haemers, Spectra of Graphs, Springer Universitext 2012. (http://homepages.cwi.nl/ aeb/math/ipm.pdf). 2 P . A. Catlin, H.-J. Lai and Y. Shao, Edge-connectivity and edge-disjoint spanning trees, Discrete Math., 309 (2009), 1033-1040. 3 S. M. Cioab˘ a and W.Wong, Edge-disjoint spanning trees and eigenvalues of regular graphs, Linear Algebra Appl., 437 (2012) 630-647. – p. 26/39

References 1 A.E. Brouwer and W.H. Haemers, Spectra of Graphs, Springer Universitext 2012. (http://homepages.cwi.nl/ aeb/math/ipm.pdf). 2 P . A. Catlin, H.-J. Lai and Y. Shao, Edge-connectivity and edge-disjoint spanning trees, Discrete Math., 309 (2009), 1033-1040. 3 S. M. Cioab˘ a and W.Wong, Edge-disjoint spanning trees and eigenvalues of regular graphs, Linear Algebra Appl., 437 (2012) 630-647. 4 W.H. Haemers, Interlacing eigenvalues and graphs, Linear Algebra Appl. 226/228 (1995), 593-616. – p. 26/39

References 1 A.E. Brouwer and W.H. Haemers, Spectra of Graphs, Springer Universitext 2012. (http://homepages.cwi.nl/ aeb/math/ipm.pdf). 2 P . A. Catlin, H.-J. Lai and Y. Shao, Edge-connectivity and edge-disjoint spanning trees, Discrete Math., 309 (2009), 1033-1040. 3 S. M. Cioab˘ a and W.Wong, Edge-disjoint spanning trees and eigenvalues of regular graphs, Linear Algebra Appl., 437 (2012) 630-647. 4 W.H. Haemers, Interlacing eigenvalues and graphs, Linear Algebra Appl. 226/228 (1995), 593-616. 5 G. Li and L. Shi, Edge-disjoint spanning trees and eigenvalues of graphs, Linear Algebra Appl. 439 (2013), 2784-2789. – p. 26/39

References 6 X. Gu, H. Lai, P . Li, S. Yao, Edge-disjoint spanning trees, edge connectivity and eigenvalues in graphs, J. Graph Theory, 81 (2016) 16-29. – p. 27/39

References 6 X. Gu, H. Lai, P . Li, S. Yao, Edge-disjoint spanning trees, edge connectivity and eigenvalues in graphs, J. Graph Theory, 81 (2016) 16-29. 7 Q. Liu, Y. Hong, H. Lai, Edge-disjoint spanning trees and eigenvalues, Linear Algebra Appl., 444 (2014) 146-151. – p. 27/39

References 6 X. Gu, H. Lai, P . Li, S. Yao, Edge-disjoint spanning trees, edge connectivity and eigenvalues in graphs, J. Graph Theory, 81 (2016) 16-29. 7 Q. Liu, Y. Hong, H. Lai, Edge-disjoint spanning trees and eigenvalues, Linear Algebra Appl., 444 (2014) 146-151. 8 Q. Liu, Y. Hong, X. Gu, H. Lai, Note on Edge-disjoint spanning trees and eigenvalues, Linear Algebra Appl., 458 (2014), 128-133. – p. 27/39

References 6 X. Gu, H. Lai, P . Li, S. Yao, Edge-disjoint spanning trees, edge connectivity and eigenvalues in graphs, J. Graph Theory, 81 (2016) 16-29. 7 Q. Liu, Y. Hong, H. Lai, Edge-disjoint spanning trees and eigenvalues, Linear Algebra Appl., 444 (2014) 146-151. 8 Q. Liu, Y. Hong, X. Gu, H. Lai, Note on Edge-disjoint spanning trees and eigenvalues, Linear Algebra Appl., 458 (2014), 128-133. 9 Y. Hong, X. Gu, H. Lai, Q. Liu, Fractional spanning tree packing, forest covering and eigenvalues, Discrete Applied Math., 213 (2016) 219-223. – p. 27/39

References 6 X. Gu, H. Lai, P . Li, S. Yao, Edge-disjoint spanning trees, edge connectivity and eigenvalues in graphs, J. Graph Theory, 81 (2016) 16-29. 7 Q. Liu, Y. Hong, H. Lai, Edge-disjoint spanning trees and eigenvalues, Linear Algebra Appl., 444 (2014) 146-151. 8 Q. Liu, Y. Hong, X. Gu, H. Lai, Note on Edge-disjoint spanning trees and eigenvalues, Linear Algebra Appl., 458 (2014), 128-133. 9 Y. Hong, X. Gu, H. Lai, Q. Liu, Fractional spanning tree packing, forest covering and eigenvalues, Discrete Applied Math., 213 (2016) 219-223. – p. 27/39

Connectivity and eigenvalue Problem (Abiad, Brimkov, Mart´ lnez-Rivera, O, and Zhang, Electronic Journal of Linear Algebra, 2018) Find best possible condition on λ 2 ( G ) to warrant κ ( G ) ≥ k . – p. 28/39

Connectivity and eigenvalue Problem (Abiad, Brimkov, Mart´ lnez-Rivera, O, and Zhang, Electronic Journal of Linear Algebra, 2018) Find best possible condition on λ 2 ( G ) to warrant κ ( G ) ≥ k . Let d and k be integers with d ≥ k ≥ 2 and G be a d -regular multigraph. Each of the following holds. – p. 28/39