Displaying similar documents to “On Laplacian eigenvalues of connected graphs”

Inequalities for real number sequences with applications in spectral graph theory

Emina Milovanović, Şerife Burcu Bozkurt Altındağ, Marjan Matejić, Igor Milovanović (2022)

Czechoslovak Mathematical Journal

Similarity:

Let a = ( a 1 , a 2 , ... , a n ) be a nonincreasing sequence of positive real numbers. Denote by S = { 1 , 2 , ... , n } the index set and by J k = { I = { r 1 , r 2 , ... , r k } , 1 r 1 < r 2 < < r k n } the set of all subsets of S of cardinality k , 1 k n - 1 . In addition, denote by a I = a r 1 + a r 2 + + a r k , 1 k n - 1 , 1 r 1 < r 2 < < r k n , the sum of k arbitrary elements of sequence a , where a I 1 = a 1 + a 2 + + a k and a I n = a n - k + 1 + a n - k + 2 + + a n . We consider bounds of the quantities R S k ( a ) = a I 1 / a I n , L S k ( a ) = a I 1 - a I n and S k , α ( a ) = I J k a I α in terms of A = i = 1 n a i and B = i = 1 n a i 2 . Then we use the obtained results to generalize some results regarding Laplacian and normalized Laplacian eigenvalues of graphs.

On the multiplicity of Laplacian eigenvalues for unicyclic graphs

Fei Wen, Qiongxiang Huang (2022)

Czechoslovak Mathematical Journal

Similarity:

Let G be a connected graph of order n and U a unicyclic graph with the same order. We firstly give a sharp bound for m G ( μ ) , the multiplicity of a Laplacian eigenvalue μ of G . As a straightforward result, m U ( 1 ) n - 2 . We then provide two graph operations (i.e., grafting and shifting) on graph G for which the value of m G ( 1 ) is nondecreasing. As applications, we get the distribution of m U ( 1 ) for unicyclic graphs on n vertices. Moreover, for the two largest possible values of m U ( 1 ) { n - 5 , n - 3 } , the corresponding graphs U are...

On the bounds of Laplacian eigenvalues of k -connected graphs

Xiaodan Chen, Yaoping Hou (2015)

Czechoslovak Mathematical Journal

Similarity:

Let μ n - 1 ( G ) be the algebraic connectivity, and let μ 1 ( G ) be the Laplacian spectral radius of a k -connected graph G with n vertices and m edges. In this paper, we prove that μ n - 1 ( G ) 2 n k 2 ( n ( n - 1 ) - 2 m ) ( n + k - 2 ) + 2 k 2 , with equality if and only if G is the complete graph K n or K n - e . Moreover, if G is non-regular, then μ 1 ( G ) < 2 Δ - 2 ( n Δ - 2 m ) k 2 2 ( n Δ - 2 m ) ( n 2 - 2 n + 2 k ) + n k 2 , where Δ stands for the maximum degree of G . Remark that in some cases, these two inequalities improve some previously known results.

Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent

Hongbin Wang (2016)

Czechoslovak Mathematical Journal

Similarity:

Let Ω L s ( S n - 1 ) for s 1 be a homogeneous function of degree zero and b a BMO function. The commutator generated by the Marcinkiewicz integral μ Ω and b is defined by [ b , μ Ω ] ( f ) ( x ) = ( 0 | x - y | t Ω ( x - y ) | x - y | n - 1 [ b ( x ) - b ( y ) ] f ( y ) d y | 2 d t t 3 1 / 2 . In this paper, the author proves the ( L p ( · ) ( n ) , L p ( · ) ( n ) ) -boundedness of the Marcinkiewicz integral operator μ Ω and its commutator [ b , μ Ω ] when p ( · ) satisfies some conditions. Moreover, the author obtains the corresponding result about μ Ω and [ b , μ Ω ] on Herz spaces with variable exponent.

Some properties of generalized distance eigenvalues of graphs

Yuzheng Ma, Yan Ling Shao (2024)

Czechoslovak Mathematical Journal

Similarity:

Let G be a simple connected graph with vertex set V ( G ) = { v 1 , v 2 , , v n } and edge set E ( G ) , and let d v i be the degree of the vertex v i . Let D ( G ) be the distance matrix and let T r ( G ) be the diagonal matrix of the vertex transmissions of G . The generalized distance matrix of G is defined as D α ( G ) = α T r ( G ) + ( 1 - α ) D ( G ) , where 0 α 1 . Let λ 1 ( D α ( G ) ) λ 2 ( D α ( G ) ) ... λ n ( D α ( G ) ) be the generalized distance eigenvalues of G , and let k be an integer with 1 k n . We denote by S k ( D α ( G ) ) = λ 1 ( D α ( G ) ) + λ 2 ( D α ( G ) ) + ... + λ k ( D α ( G ) ) the sum of the k largest generalized distance eigenvalues. The generalized distance spread of a graph G is defined as D α S ( G ) = λ 1 ( D α ( G ) ) - λ n ( D α ( G ) ) ....

On the signless Laplacian and normalized signless Laplacian spreads of graphs

Emina Milovanović, Serife B. Bozkurt Altindağ, Marjan Matejić, Igor Milovanović (2023)

Czechoslovak Mathematical Journal

Similarity:

Let G = ( V , E ) , V = { v 1 , v 2 , ... , v n } , be a simple connected graph with n vertices, m edges and a sequence of vertex degrees d 1 d 2 d n . Denote by A and D the adjacency matrix and diagonal vertex degree matrix of G , respectively. The signless Laplacian of G is defined as L + = D + A and the normalized signless Laplacian matrix as + = D - 1 / 2 L + D - 1 / 2 . The normalized signless Laplacian spreads of a connected nonbipartite graph G are defined as r ( G ) = γ 2 + / γ n + and l ( G ) = γ 2 + - γ n + , where γ 1 + γ 2 + γ n + 0 are eigenvalues of + . We establish sharp lower and upper bounds for the normalized signless...

Almost everywhere convergence of convolution powers on compact abelian groups

Jean-Pierre Conze, Michael Lin (2013)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

It is well-known that a probability measure μ on the circle 𝕋 satisfies μ n * f - f d m p 0 for every f L p , every (some) p [ 1 , ) , if and only if | μ ^ ( n ) | l t ; 1 for every non-zero n ( μ is strictly aperiodic). In this paper we study the a.e. convergence of μ n * f for every f L p whenever p g t ; 1 . We prove a necessary and sufficient condition, in terms of the Fourier–Stieltjes coefficients of μ , for the strong sweeping out property (existence of a Borel set B with lim sup μ n * 1 B = 1 a.e. and lim inf μ n * 1 B = 0 a.e.). The results are extended to general compact Abelian groups...

Radon-Nikodym property

Surjit Singh Khurana (2017)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

For a Banach space E and a probability space ( X , 𝒜 , λ ) , a new proof is given that a measure μ : 𝒜 E , with μ λ , has RN derivative with respect to λ iff there is a compact or a weakly compact C E such that | μ | C : 𝒜 [ 0 , ] is a finite valued countably additive measure. Here we define | μ | C ( A ) = sup { k | μ ( A k ) , f k | } where { A k } is a finite disjoint collection of elements from 𝒜 , each contained in A , and { f k } E ' satisfies sup k | f k ( C ) | 1 . Then the result is extended to the case when E is a Frechet space.