New bounds on the Laplacian spectral ratio of connected graphs

Zhen Lin; Min Cai; Jiajia Wang

Displaying similar documents to “New bounds on the Laplacian spectral ratio of connected graphs”

Spectral projections for the twisted Laplacian

Herbert Koch, Fulvio Ricci (2007)

Studia Mathematica

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Let n ≥ 1, d = 2n, and let (x,y) ∈ ℝⁿ × ℝⁿ be a generic point in ℝ²ⁿ. The twisted Laplacian $L = - 1 / 2 \sum_{j = 1}^{n} [(\partial_{x_{j}} + i y_{j}) ² + (\partial_{y_{j}} - i x_{j}) ²]$ has the spectrum n + 2k = λ²: k a nonnegative integer. Let $P_{λ}$ be the spectral projection onto the (infinite-dimensional) eigenspace. We find the optimal exponent ϱ(p) in the estimate $| | P_{λ} {u | |}_{L^{p} (ℝ^{d})} ≲ λ^{ϱ (p)} {| | u | |}_{L ² (ℝ^{d})}$ for all p ∈ [2,∞], improving previous partial results by Ratnakumar, Rawat and Thangavelu, and by Stempak and Zienkiewicz. The expression for ϱ(p) is ϱ(p) = 1/p -1/2 if 2 ≤ p ≤ 2(d+1)/(d-1), ϱ(p) = (d-2)/2 - d/p...

Sufficient conditions on the existence of factors in graphs involving minimum degree

Huicai Jia, Jing Lou (2024)

Czechoslovak Mathematical Journal

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For a set ${A, B, C, ...}$ of graphs, an ${A, B, C, ...}$ -factor of a graph $G$ is a spanning subgraph $F$ of $G$ , where each component of $F$ is contained in ${A, B, C, ...}$ . It is very interesting to investigate the existence of factors in a graph with given minimum degree from the prospective of eigenvalues. We first propose a tight sufficient condition in terms of the $Q$ -spectral radius for a graph involving minimum degree to contain a star factor. Moreover, we also present tight sufficient conditions based on the $Q$ -spectral radius...

A spectral bound for graph irregularity

Felix Goldberg (2015)

Czechoslovak Mathematical Journal

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The imbalance of an edge $e = {u, v}$ in a graph is defined as $i (e) = | d (u) - d (v) |$ , where $d (\cdot)$ is the vertex degree. The irregularity $I (G)$ of $G$ is then defined as the sum of imbalances over all edges of $G$ . This concept was introduced by Albertson who proved that $I (G) \leq 4 n^{3} / 27$ (where $n = | V (G) |$ ) and obtained stronger bounds for bipartite and triangle-free graphs. Since then a number of additional bounds were given by various authors. In this paper we prove a new upper bound, which improves a bound found by Zhou and Luo in 2008. Our bound involves...

Unbalanced unicyclic and bicyclic graphs with extremal spectral radius

Francesco Belardo, Maurizio Brunetti, Adriana Ciampella (2021)

Czechoslovak Mathematical Journal

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A signed graph $Γ$ is a graph whose edges are labeled by signs. If $Γ$ has $n$ vertices, its spectral radius is the number $ρ (Γ) : = max {| λ_{i} (Γ) | : 1 \leq i \leq n}$ , where $λ_{1} (Γ) \geq \dots \geq λ_{n} (Γ)$ are the eigenvalues of the signed adjacency matrix $A (Γ)$ . Here we determine the signed graphs achieving the minimal or the maximal spectral radius in the classes $𝔘_{n}$ and $𝔅_{n}$ of unbalanced unicyclic graphs and unbalanced bicyclic graphs, respectively.

On distance Laplacian energy in terms of graph invariants

Hilal A. Ganie, Rezwan Ul Shaban, Bilal A. Rather, Shariefuddin Pirzada (2023)

Czechoslovak Mathematical Journal

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For a simple connected graph $G$ of order $n$ having distance Laplacian eigenvalues $ρ_{1}^{L} \geq ρ_{2}^{L} \geq \dots \geq ρ_{n}^{L}$ , the distance Laplacian energy $DLE (G)$ is defined as $DLE (G) = \sum_{i = 1}^{n} | ρ_{i}^{L} - 2 W (G) / n |$ , where $W (G)$ is the Wiener index of $G$ . We obtain a relationship between the Laplacian energy and the distance Laplacian energy for graphs with diameter 2. We obtain lower bounds for the distance Laplacian energy $DLE (G)$ in terms of the order $n$ , the Wiener index $W (G)$ , the independence number, the vertex connectivity number and other given parameters. We characterize the...

On the spectral radius in $L_{1} (G)$

E. Porada (1971)

Colloquium Mathematicae

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On the spectrum of the operator which is a composition of integration and substitution

Ignat Domanov (2008)

Studia Mathematica

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Let ϕ: [0,1] → [0,1] be a nondecreasing continuous function such that ϕ(x) > x for all x ∈ (0,1). Let the operator $V_{ϕ} : f (x) \mapsto \int_{0}^{ϕ (x)} f (t) d t$ be defined on L₂[0,1]. We prove that $V_{ϕ}$ has a finite number of nonzero eigenvalues if and only if ϕ(0) > 0 and ϕ(1-ε) = 1 for some 0 < ε < 1. Also, we show that the spectral trace of the operator $V_{ϕ}$ always equals 1.

On the bounds of Laplacian eigenvalues of $k$ -connected graphs

Xiaodan Chen, Yaoping Hou (2015)

Czechoslovak Mathematical Journal

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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 $μ_{n - 1} (G) \geq \frac{2 n k^{2}}{(n (n - 1) - 2 m) (n + k - 2) + 2 k^{2}},$ with equality if and only if $G$ is the complete graph $K_{n}$ or $K_{n} - e$ . Moreover, if $G$ is non-regular, then $μ_{1} (G) < 2 Δ - \frac{2 (n Δ - 2 m) k^{2}}{2 (n Δ - 2 m) (n^{2} - 2 n + 2 k) + n k^{2}},$ where $Δ$ stands for the maximum degree of $G$ . Remark that in some cases, these two inequalities improve some previously known results.

On the multiplicity of Laplacian eigenvalues for unicyclic graphs

Fei Wen, Qiongxiang Huang (2022)

Czechoslovak Mathematical Journal

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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, $m_{U} (1) \leq n - 2$ . 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 $m_{U} (1) \in {n - 5, n - 3}$ , the corresponding graphs $U$ are...

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

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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 $d_{1} \geq d_{2} \geq \dots \geq d_{n}$ . 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 $ℒ^{+} = D^{- 1 / 2} L^{+} D^{- 1 / 2}$ . 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 $γ_{1}^{+} \geq γ_{2}^{+} \geq \dots \geq γ_{n}^{+} \geq 0$ are eigenvalues of $ℒ^{+}$ . We establish sharp lower and upper bounds for the normalized signless...

The spectral determinations of the connected multicone graphs $K_{w} ▽ m P_{17}$ and $K_{w} ▽ m S$

Ali Zeydi Abdian, S. Morteza Mirafzal (2018)

Czechoslovak Mathematical Journal

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Finding and discovering any class of graphs which are determined by their spectra is always an important and interesting problem in the spectral graph theory. The main aim of this study is to characterize two classes of multicone graphs which are determined by both their adjacency and Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let $K_{w}$ denote a complete graph on $w$ vertices, and let $m$ be a positive integer number. In A. Z. Abdian (2016)...

Partial sum of eigenvalues of random graphs

Israel Rocha (2020)

Applications of Mathematics

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Let $G$ be a graph on $n$ vertices and let $λ_{1} \geq λ_{2} \geq ... \geq λ_{n}$ be the eigenvalues of its adjacency matrix. For random graphs we investigate the sum of eigenvalues $s_{k} = \sum_{i = 1}^{k} λ_{i}$ , for $1 \leq k \leq n$ , and show that a typical graph has $s_{k} \leq (e (G) + k^{2}) / {(0.99 n)}^{1 / 2}$ , where $e (G)$ is the number of edges of $G$ . We also show bounds for the sum of eigenvalues within a given range in terms of the number of edges. The approach for the proofs was first used in Rocha (2020) to bound the partial sum of eigenvalues of the Laplacian matrix.

Spectral synthesis and operator synthesis

K. Parthasarathy, R. Prakash (2006)

Studia Mathematica

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Relations between spectral synthesis in the Fourier algebra A(G) of a compact group G and the concept of operator synthesis due to Arveson have been studied in the literature. For an A(G)-submodule X of VN(G), X-synthesis in A(G) has been introduced by E. Kaniuth and A. Lau and studied recently by the present authors. To any such X we associate a $V^{\infty} (G)$ -submodule X̂ of ℬ(L²(G)) (where $V^{\infty} (G)$ is the weak-* Haagerup tensor product $L^{\infty} (G) \otimes_{w * h} L^{\infty} (G)$ ), define the concept of X̂-operator synthesis and prove that a...

A.e. convergence of anisotropic partial Fourier integrals on Euclidean spaces and Heisenberg groups

D. Müller, E. Prestini (2010)

Colloquium Mathematicae

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We define partial spectral integrals $S_{R}$ on the Heisenberg group by means of localizations to isotropic or anisotropic dilates of suitable star-shaped subsets V containing the joint spectrum of the partial sub-Laplacians and the central derivative. Under the assumption that an L²-function f lies in the logarithmic Sobolev space given by $l o g (2 + L_{α}) f \in L ²$ , where $L_{α}$ is a suitable “generalized” sub-Laplacian associated to the dilation structure, we show that $S_{R} f (x)$ converges a.e. to f(x) as R → ∞.

Behaviour of the first eigenvalue of the p-Laplacian in a domain with a hole

M. Sango (2001)

Colloquium Mathematicae

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We investigate the behaviour of a sequence $λ_{s}$ , s = 1,2,..., of eigenvalues of the Dirichlet problem for the p-Laplacian in the domains $Ω_{s}$ , s = 1,2,..., obtained by removing from a given domain Ω a set $E_{s}$ whose diameter vanishes when s → ∞. We estimate the deviation of $λ_{s}$ from the eigenvalue of the limit problem. For the derivation of our results we construct an appropriate asymptotic expansion for the sequence of solutions of the original eigenvalue problem.

A spectral gap theorem in SU $(d)$

Jean Bourgain, Alex Gamburd (2012)

Journal of the European Mathematical Society

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We establish the spectral gap property for dense subgroups of SU $(d)$ $(d \geq 2)$ , generated by finitely many elements with algebraic entries; this result was announced in [BG3]. The method of proof differs, in several crucial aspects, from that used in [BG] in the case of SU $(2)$ .

Transferring $L^{p}$ eigenfunction bounds from $S^{2 n + 1}$ to hⁿ

Valentina Casarino, Paolo Ciatti (2009)

Studia Mathematica

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By using the notion of contraction of Lie groups, we transfer $L^{p} - L ²$ estimates for joint spectral projectors from the unit complex sphere $S^{2 n + 1}$ in $ℂ^{n + 1}$ to the reduced Heisenberg group hⁿ. In particular, we deduce some estimates recently obtained by H. Koch and F. Ricci on hⁿ. As a consequence, we prove, in the spirit of Sogge’s work, a discrete restriction theorem for the sub-Laplacian L on hⁿ.