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Let G be a graph of order n such that ∑n i=0(-1) iailambdan-i and ∑n i=0(-1) iailambdan-i are the characteristic polynomials of the signless Laplacian and the Laplacian matrices of G, respectively. We show that a i ≥b i for i=0,1,⋯,n. As a consequence, we prove that for any α, 0<α≤1, if q 1,⋯,q n and μ 1,⋯,μ n are the signless Laplacian and the Laplacian eigenvalues of G, respectively, then q 1 alpha+⋯+qα n≥μ α 1+⋯+μα n  

For a graph G, let λ2(G) denote its second smallest Laplacian eigenvalue. It was conjectured that (Formula presented.), where (Formula presented.) is the complement of G. In this paper, it is shown that (Formula presented.). © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group  

For a graph G, let λ2(G) denote its second smallest Laplacian eigenvalue. It was conjectured that (Formula presented.), where (Formula presented.) is the complement of G. In this paper, it is shown that (Formula presented.). © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group  

For a graph G, let (Formula presented.) denote its second smallest Laplacian eigenvalue. It was conjectured that (Formula presented.), where (Formula presented.) is the complement of G. In this paper, it is shown that (Formula presented.). © 2019 Informa UK Limited, trading as Taylor & Francis Group  

For a graph G, let (Formula presented.) denote its second smallest Laplacian eigenvalue. It was conjectured that (Formula presented.), where (Formula presented.) is the complement of G. In this paper, it is shown that (Formula presented.). © 2019 Informa UK Limited, trading as Taylor & Francis Group  

It is well known that all starlike trees, i.e. trees with exactly one vertex of degree at least three, are determined by their Laplacian spectrum. A double starlike tree is a tree with exactly two vertices of degree at least three. In 2009, the following question was posed: Are all the double starlike trees determined by their Laplacian spectra? In this direction, it was proved that one special double starlike tree Hn(p,p) is determined by its Laplacian spectrum, where Hn(p,q) is a tree obtained by joining p pendant vertices to an end vertex of a path of order and then joining pendant vertices to another end of the path. Also, the banana tree Bn,k is a tree obtained by joining a vertex to... 

For a graph G, let λ2(G) denote its second smallest Laplacian eigenvalue. It was conjectured that λ2(G)+λ2(G‾)≥1, where G‾ is the complement of G. In this paper, it is shown that max⁡{λ2(G),λ2(G‾)}≥[Formula presented]. © 2018 Elsevier Inc  

For a graph G of order n, let λ 2 (G) denote its second smallest Laplacian eigenvalue. It was conjectured that λ 2 (G)+λ 2 (G‾)≥1, where G‾ is the complement of G. For any x∈R n , let ∇ x ∈R (n2) be the vector whose {i,j}-th entry is |x i −x j |. In this paper, we show the aforementioned conjecture is equivalent to prove that every two orthonormal vectors f,g∈R n with zero mean satisfy ‖∇ f −∇ g ‖ 2 ≥2. In this article, it is shown that for the validity of the conjecture it suffices to prove that the conjecture holds for all permutation graphs. © 2019 Elsevier Inc  

Embedding metrics into constant-dimensional geometric spaces, such as the Euclidean plane, is relatively poorly understood. Motivated by applications in visualization, ad-hoc networks, and molecular reconstruction, we consider the natural problem of embedding shortest-path metrics of unweighted planar graphs (planar graph metrics) into the Euclidean plane. It is known that, in the special case of shortest-path metrics of trees, embedding into the plane requires Θ(√n) distortion in the worst case [M1], [BMMV], and surprisingly, this worst-case upper bound provides the best known approximation algorithm for minimizing distortion. We answer an open question posed in this work and highlighted by... 

Let G be a graph and L(G) be the Laplacian matrix of G. In this paper, we explicitly determine the multiplicity of Laplacian eigenvalue 2 for any unicyclic graph containing a perfect matching  

Metric learning is a powerful approach for semi-supervised clustering. In this paper, a metric learning method considering both pairwise constraints and the geometrical structure of data is introduced for semi-supervised clustering. At first, a smooth metric is found (based on an optimization problem) using positive constraints as supervisory information. Then, an extension of this method employing both positive and negative constraints is introduced. As opposed to the existing methods, the extended method has the capability of considering both positive and negative constraints while considering the topological structure of data. The proposed metric learning method can improve performance of... 

Let G be a simple connected graph of order n. Let (Formula presented.) be the Laplacian eigenvalues of G. In this paper, we show that if X and Y are two subsets of vertices of G such that (Formula presented.) and the set of all edges between X and Y decomposed into r disjoint perfect matchings, then, (Formula presented.) where (Formula presented.). Also, we determine a relation between the Laplacian eigenvalues and matchings in a bipartite graph by showing that if (Formula presented.) is a bipartite graph, (Formula presented.) and (Formula presented.), then G has a matching that saturates U. © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group  

Let G be a simple graph with n vertices and let λ1, λ2,...,λn be its eigenvalues. The energy of G is defined to be E(G) = ∑i=1n|λi|. In this note, for a given k-regular graph we find explicit formulas for the energy of L(G), the complement of line graph of G. This provides us with some practical ways to compute the energy of a large family of regular graphs  

The energy E of any n-vertex regular graph G of degree r, r > 0, is greater than or equal to n. Equality holds if and only if every component of G is isomorphic to the complete bipartite graph Kr,r. If G is triangle- and quadrangle-free, then E ≥ nr/√2r - 1. In particular, for any fullerene and nanotube with n carbon atoms, 1.34n ≤ E ≤ 1.73 n  

We study a class of compactifications of open complex analytic surfaces which appear naturally in the study of the singularities of Calabi-Yau metrics. We obtain a new degenerate Hodge theory as well as a degenerate Dolbeault lemma for these surfaces. To cite this article: A. Bahraini, C. R. Acad. Sci. Paris, Ser. I 344 (2007). © 2007 Académie des sciences  

In this paper, we study the multiplicity of the Laplacian eigenvalues of trees. It is known that for trees, integer Laplacian eigenvalues larger than 1 are simple and also the multiplicity of Laplacian eigenvalue 1 has been well studied before. Here we consider the multiplicities of the other (non-integral) Laplacian eigenvalues. We give an upper bound and determine the trees of order n that have a multiplicity that is close to the upper bound [Formula presented], and emphasize the particular role of the algebraic connectivity. © 2019 Elsevier Inc  

In this paper, we find upper bounds for the eigenvalues of the Laplacian in the conformal class of a compact Riemannian manifold (M,g). These upper bounds depend only on the dimension and a conformal invariant that we call "min-conformal volume". Asymptotically, these bounds are consistent with the Weyl law and improve previous results by Korevaar and Yang and Yau. The proof relies on the construction of a suitable family of disjoint domains providing supports for a family of test functions. This method is interesting for itself and powerful. As a further application of the method we obtain an upper bound for the eigenvalues of the Steklov problem in a domain with C1 boundary in a complete... 

In this paper we state and prove delocalized Morse type inequalities for Morse functions as well as for closed differential 1-forms. These inequalities involve delocalized Betti numbers. As an immediate consequence, we prove the vanishing of delocalized Betti numbers of manifolds fibering over the circle under a vanishing condition on the delocalizing conjugacy class  

In this paper, we study the existence of a positive solution to the elliptic problem: (Formula presented.) Here (Formula presented.) (N>2s) is an open bounded domain with smooth boundary, (Formula presented.) and (Formula presented.). For (Formula presented.), we take advantage of the convexity of Ω. The operator (Formula presented.) indicates the restricted fractional Laplacian, and μ is a non-negative Radon measure as a source term. The assumptions on f and h will be precised later. Besides, we will discuss the notion of entropy solution and its uniqueness for some specific measures. © 2020, © 2020 Informa UK Limited, trading as Taylor & Francis Group