Some aspects of spectral graph theory u H July 2018 - PowerPoint PPT Presentation

Some aspects of spectral graph theory ± Å u H “ ‰ Œ Æ ê Æ ‰ ÆÆ � July 2018

Introduction Let G be a graph with vertex set V ( G ) and edge set E ( G ), where n = | V ( G ) | is the order and m = | E ( G ) | is the size of G . d G ( u ): the degree of vertex u in G .

Introduction A ( G ): the (0 , 1) adjacency matrix of G . L ( G ): the Laplacian matrix of G . Q ( G ): the signless Laplacian matrix of G . D ( G ): the distance matrix of (connected) G . ρ 1 ( G ) ≥ · · · ≥ ρ n ( G ) = ρ min ( G ): the (adjacency) eigenvalues of G . λ 1 ( G ) ≥ · · · ≥ λ n ( G ): the Laplacian eigenvalues of G . µ 1 ( G ) ≥ · · · ≥ µ n ( G ): the signless Laplacian eigenvalues of G . γ 1 ( G ) ≥ · · · ≥ γ n ( G ): the distance eigenvalues of G .

(Adjacency) eigenvalues A classical result (A.J. Hoffman, 1970) is: for a nonempty graph G on n vertices, χ ( G ) ≥ 1 + ρ 1 . − ρ n Another one is χ ( G ) ≤ 1 + ρ 1 with equality if and only if G is a complete graph or an odd cycle. N.L. Biggs, Algebraic graph theory, Cambridge Univ. Press, Cambridge, 1974. H.S. Wilf, The eigenvalues of a graph and its chromatic number, J. London Math. Soc. 42 (1967) 330–332. P. Wocjan, C. Elphick, New spectral bounds on the chromatic number encompassing all eigenvalues of the adjacency matrix, Electron. J. Combin. 20 (2013) Paper 39.

Least (adjacency) eigenvalue A connected graph is a cactus if any two of its cycles share at most one common vertex. Petrovi´ c, Aleksi´ c and Simi´ c (2011) determined the unique cactus whose least eigenvalue is minimal among the cacti with n vertices and k cycles, where 0 ≤ k ≤ ⌊ n − 1 2 ⌋ . T n , p : the tree consisting of p pendant paths at a common vertex with almost equal lengths.

Least (adjacency) eigenvalue C n , k ( r , s ; a ) with 2 r + 3 s − a = n − 2 k − 1, 1 ≤ k ≤ n − 3, and if r ≥ 1, then a = k . s r � �� � s s � �� � ✁ ✁❆ ❆ ✜ ✜ ❭ ❭ ◗ ✑ s s s♣ ♣ ♣ s ❈ s ☎ s s ♣ ♣ ♣ s ◗ ❅ ❈ ✑✑✑✑ ☎ � ❆ ✁✁ ◗ ❅ ❆ ❈ ☎ � ◗ ◗ ❅ ❆ ❈ ☎ ✁ � s v 0 ✑ ◗◗◗ ☞ ▲ ✑ ✑ ☞ ▲ ✑ ◗ s ☞ ▲ s s q q q s � �� � a s q q q s � �� � k − a The graph C n , k ( r , s ; a ).

Least (adjacency) eigenvalue Theorem (Xing, Z) Let G be a graph with minimum least eigenvalue among the n -vertex cacti with k pendant vertices, where 1 ≤ k ≤ n − 1. Let n 0 ( k ) = 18 for 1 ≤ k ≤ 3, n 0 ( k ) = 17 for 4 ≤ k ≤ 7, and n 0 ( k ) = 16 for 8 ≤ k ≤ 10. Then for k = n − 1 , n − 2, G ∼ = T n , k , and for 1 ≤ k ≤ n − 3, we have (i) if 1 ≤ k ≤ 10 and n < n 0 ( k ), then � � (a) for n − k ≡ 0 ( mod 3), G ∼ 1 , n − k − 3 ; k , = C n , k 3 � � (b) for n − k ≡ 1 ( mod 3), G ∼ 0 , n − k − 1 = C n , k ; k , 3 (c) for n − k ≡ 2 ( mod 3), G ∼ = C n , k (0 , n − k − 2 ; k − 1) if 3 ( n , k ) = (6 , 1), (9 , 1), (7 , 2), (8 , 3), or (9 , 4), and � � G ∼ 2 , n − k − 5 = C n , k ; k otherwise; 3 (ii) if 1 ≤ k ≤ 10 and n ≥ n 0 ( k ), or k ≥ 11, then � n − k − 4 � (a) for n − k ≡ 0 ( mod 2), G ∼ = C n , k , 1; k , 2 � n − k − 1 � (b) for n − k ≡ 1 ( mod 2), G ∼ , 0; k . = C n , k 2

Least (adjacency) eigenvalue Theorem (Liu, Z) Let G be a graph with minimum least eigenvalue among the n -vertex bicyclic graphs with k pendant vertices, where 1 ≤ k ≤ n − 1. Then . . . .

(Adjacency) eigenvalues For a graph G with U ⊂ V ( G ) and F ⊆ E ( G ), Li et al. gave a new type lower bound for ρ 1 ( G − U ) in terms of ρ 1 ( G ) and the entries of the Perron vector of G . Mieghem et al. gave lower and upper bounds for ρ 1 ( G − F ) in terms of ρ 1 ( G ) and the entries of the Perron vector of G and G − F . C. Li, H. Wang, P. Van Mieghem, Bounds for the spectral radius of a graph when nodes are removed, Linear Algebra Appl. 437 (2012) 319 õ 323. P. Van Mieghem, D. Stevanovi´ c, F.A. Kuipers, C. Li, R. van de Bovenkamp, D. Liu, H. Wang, Decreasing the spectral radius of a graph by link removals, Phys. Rev. E 84 (1) (2011) 016101

Least (adjacency) eigenvalue Theorem (Xing, Z) Let G be a graph with x being a least eigenvector. For U ⊂ V ( G ), we have � � � � x 2 ρ min ( G ) ≤ ρ min ( G − U ) ≤ 1 − 2 ρ min ( G ) + 2 x i x j . i i ∈ U { i , j }⊆ U i ∼ j In particular, ρ min ( G ) ≤ ρ min ( G − i ) ≤ (1 − 2 x 2 i ) ρ min ( G ), where i ∈ V ( G ).

Least (adjacency) eigenvalue Theorem (Xing, Z) Let G be a graph with F ⊆ E ( G ). Let x and y be least eigenvectors of G and G − F , respectively. Then � � 2 x i x j ≤ ρ min ( G ) − ρ min ( G − F ) ≤ 2 y i y j . ij ∈ F ij ∈ F

(Adjacency) eigenvalues — Estrada index The Estrada index of a graph G is defined as n � EE ( G ) = e ρ i . i =1 J.A. de la Pe˜ na, I. Gutman, J. Rada, Estimating the Estrada index, Linear Algebra Appl. 427 (2007) 70–76. Let m k ( G ) = � n i =1 ρ k i (number of closed walks of length k in G ). Then EE ( G ) = � ∞ M k ( G ) . k =0 k ! m 0 ( G ) = n , m 1 ( G ) = 0, m 2 ( G ) = 2 m , m 3 ( G ) = 2 t .

(Adjacency) eigenvalues — Estrada index Let G and H be two graphs of order n . For integer k ≥ 2, let W k ( G ) be the number of closed walk of length k in G . If we can establish a injection σ k from W k ( G ) to W k ( H ) for all k , then M k ( G ) ≤ M k ( H ), implying that EE ( G ) ≤ EE ( H ). Moreover, if for some k 0 , σ k 0 is not a surjection, EE ( G ) < EE ( H ). M k 0 ( G ) < M k 0 ( H ).

(Adjacency) eigenvalues — Estrada index Theorem (Du,Z) Let G be a tree with n vertices and p pendant vertices, where 2 ≤ p ≤ n − 1. Then EE ( G ) ≤ EE ( T n , p ) with equality if and only if G ∼ = T n , p . For 2 ≤ r ≤ ⌊ n / 2 ⌋ , let T n , r be the tree obtained by attaching r − 1 paths on two vertices to the center of the star S n − 2 r +2 . Corollary Let G be a tree with n vertices and matching number m , where 2 ≤ m ≤ ⌊ n / 2 ⌋ . Then EE ( G ) ≤ EE ( T n , m ) with equality if and only if G ∼ = T n , m . Let G be a tree with n vertices and independence number α , where ⌈ n / 2 ⌉ ≤ α ≤ n − 2. Then EE ( G ) ≤ EE ( T n , n − α ) with equality if and only if G ∼ = T n , n − α . Let G be a tree with n vertices and domination number γ , where 2 ≤ γ ≤ ⌊ n / 2 ⌋ . Then EE ( G ) ≤ EE ( T n ,γ ) with equality if and only if G ∼ = T n ,γ .

(Adjacency) eigenvalues — Estrada index Let D n , ∆ be the tree obtained by adding an edge between the centers of two vertex-disjoint stars S ∆ , and attaching a path on n − 2∆ vertices to a pendant vertex, where n ≥ 2∆ + 1 ≥ 7 [MATCH Commun. Math. Comput. Chem. 64 (2010) 799–810]. Theorem (Du,Z) Let G be an n -vertex tree with two adjacent vertices of maximum degree ∆, where n ≥ 2∆ + 1 ≥ 7. Then EE ( G ) ≥ EE ( D n , ∆ ) with equality if and only if G ∼ = D n , ∆ .

(Adjacency) eigenvalues — Estrada index Theorem (Du, Z) Let G be a connected graph with n vertices and k cut edges, where 0 ≤ k ≤ n − 3. Then EE ( G ) ≤ EE ( G n , k ) with equality if and only if G ∼ = G n , K , where G n , k is the graph obtained from the complete graph on n − k vertices by attaching k pendant edges to a vertex.

(Adjacency) eigenvalues — Estrada index Theorem (Du, Z) (i) Let G be a unicyclic graph on n ≥ 4 vertices. Then EE ( G ) ≤ EE ( C 3 ( n − 3)) with equality if and only if G ∼ = C 3 ( n − 3). (ii) Let G be an n -vertex unicyclic graph, where n ≥ 5. If G �∼ = C n , H n , then EE ( G ) > min { EE ( C n ) , EE ( H n ) } .

Laplacian eigenvalues — Sum of the first k Laplacian eigenvalues Let S k ( G ) = µ 1 + · · · + µ k for a graph G with n vertices. Let d ∗ i ( G ) = |{ v ∈ V ( G ) : d v ≥ i }| for i = 1 , 2 , . . . , n . Grone–Merris conjecture (proven by Bai): k � d ∗ S k ( G ) ≤ i ( G ) for 1 ≤ k ≤ n . i =1 R. Grone, R. Merris, The Laplacian spectrum of a graph II, SIAM J. Discrete Math. 7 (1994) 221–229. H. Bai, The Grone-Merris conjecture, Trans. Amer. Math. Soc. 363 (2011) 4463–4474.

Laplacian eigenvalues — Sum of the first k Laplacian eigenvalues As a variation of the Grone–Merris conjecture, Brouwer proposed the following Brouwer conjecture: � k + 1 � S k ( G ) ≤ e ( G ) + for 1 ≤ k ≤ n , 2 where e ( G ) is the number of edges of G . A.E. Brouwer, Spectra of graphs, Springer, New York, 2012.

Laplacian eigenvalues — Sum of the first k Laplacian eigenvalues Brouwer: it is true for graphs with at most 10 vertices. For k = n − 1 or n , it follows trivially because S k ( G ) = 2 e ( G ). For k = 1, it follows from the well-known inequality µ 1 ( G ) ≤ n . Haemers et al. : it is true for all graphs when k = 2. Haemers et al. : it is true for trees. W.H. Haemers, A. Mohammadian, B. Tayfeh-Rezaie, On the sum of Laplacian eigenvalues of graphs, Linear Algebra Appl. 432 (2010) 2214–2221. Theorem Brouwer conjecture is true for unicyclic graphs and bicyclic graphs.