Let G be a simple connected graph with order n. $$\mathcal{L}(G)$$ and $$\mathcal{Q}(G)$$ the normalized Laplacian signless matrices of G, respectively. λk(G) k-th smallest eigenvalue G. Denote by ρ(A) spectral radius matrix A. In this paper, we study behaviors λ2(G) $$\rho(\mathcal{L}(G))$$ when is perturbed three operations. We also properties X for bipartite graphs, where unit eigenvector co...

Denote by Tn,q the set of trees with n vertices and matching number q. Guo [On the Laplacian spectral radius of a tree, Linear Algebra Appl. 368 (2003) 379–385] gave the tree in Tn,q with the greatest value of the largest Laplacian eigenvalue. In this paper, we give another proof of this result. Using our method, we can go further beyond Guo by giving the tree in Tn,q with the second largest va...

The main results of this paper are: 1) a proof that a necessary condition for 1 to be an eigenvalue of the S-matrix is real analyticity of the boundary of the obstacle, 2) a short proof of the conclusion stating that if 1 is an eigenvalue of the Smatrix, then k2 is an eigenvalue of the Laplacian of the interior problem, and that in this case there exists a solution to the interior Dirichlet pro...

We show that the k-th eigenvalue of the Dirichlet Laplacian is strictly less than the k-th eigenvalue of the classical Stokes operator (equivalently, of the clamped buckling plate problem) for a bounded domain in the plane having a locally Lipschitz boundary. For a C boundary, we show that eigenvalues of the Stokes operator with Navier slip (friction) boundary conditions interpolate continuousl...

This paper develops the necessary tools to understand the relationship between eigenvalues of the Laplacian matrix of a graph and the connectedness of the graph. First we prove that a graph has k connected components if and only if the algebraic multiplicity of eigenvalue 0 for the graph’s Laplacian matrix is k. We then prove Cheeger’s inequality (for dregular graphs) which bounds the number of...

For eigenvalue problems of self-adjoint differential operators, a universal framework is proposed to give explicit lower and upper bounds for the eigenvalues. In the case of the Laplacian operator, by applying Crouzeix–Raviart finite elements, an efficient algorithm is developed to bound the eigenvalues for the Laplacian defined in 1D, 2D and 3D spaces. Moreover, for nonconvex domains, for whic...

In this article, we study eigenvalue problems with the p-Laplacian operator: −(|y′|p−2y′)′ = (p− 1)(λρ(x)− q(x))|y|p−2y on (0, πp), where p > 1 and πp ≡ 2π/(p sin(π/p)). We show that if ρ ≡ 1 and q is singlewell with transition point a = πp/2, then the second Neumann eigenvalue is greater than or equal to the first Dirichlet eigenvalue; the equality holds if and only if q is constant. The same ...

Scales in RNA, based on geometrical considerations, can be exploited for the analysis and prediction of RNA structures. By using spectral decomposition, geometric information that relates to a given RNA fold can be reduced to a single positive scalar number, the second eigenvalue of the Laplacian matrix corresponding to the tree-graph representation of the RNA secondary structure. Along with th...

Let U(n, k) be the set of non-bipartite unicyclic graphs with n vertices and k pendant vertices, where n ≥ 4. In this paper, the unique graph with the minimal least eigenvalue of the signless Laplacian among all graphs in U(n, k) is determined. Furthermore, it is proved that the minimal least eigenvalue of the signless Laplacian is an increasing function in k. Let Un denote the set of non-bipar...