Graphs with maximal signless Laplacian spectral radius

By the signless Laplacian of a (simple) graph G we mean the matrix Q(G)=D(G)+A(G), where A(G),D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. It is known that connected graphs G that maximize the signless Laplacian spectral radius ρ(Q(G)) over all connected graphs with given numbers of vertices and edges are (degree) maximal. For a maximal graph G with n vertices and r distinct vertex degrees δ_r>δ_r-1>?>δ₁, it is proved that ρ(Q(G))<ρ(Q(H)) for some maximal graph H with n+1 (respectively, n) vertices and the same number of edges as G if either G has precisely two dominating vertices or there exists an integer such that δ_i+δ_r+1-i?n+1 (respectively, δ_i+δ_r+1-i?δ_l+δ_r-l+1). Graphs that maximize ρ(Q(G)) over the class of graphs with m edges and m-k vertices, for k=0,1,2,3, are completely determined.     

Real-root property of the spectral polynomial of the Treibich–Verdier potential and related problems

We study the spectral polynomial of the Treibich–Verdier potential. Such spectral polynomial, which is a generalization of the classical Lamé polynomial, plays fundamental roles in both the finite-gap theory and the ODE theory of Heun's equation. In this paper, we prove that all the roots of such spectral polynomial are real and distinct under some assumptions. The proof uses the classical concept of Sturm sequence and isomonodromic theories. We also prove an analogous result for a polynomial associated with a generalized Lamé equation, where we apply a new approach based on the viewpoint of the monodromy data.     

A modified axisymmetric finite element for the 3-D vibration analysis of piezoelectric laminated circular and annular plates

A finite element is presented to analyze the three-dimensional (3-D) vibration of piezoelectric coupled circular and annular plates. The proposed finite element is a modification of a conventional axisymmetric finite element and is capable of conducting both axisymmetric and nonaxisymmetric vibration analysis of circular and annular laminated plates, with piezoelectric layers therein. The present formulation, a two-dimensional model itself, can investigate 3-D vibration of those plates for a preselected number of nodal diameters, and is therefore more economical than the conventional 3-D finite element analysis, yet still has almost the same accuracy and versatility as the 3-D analysis. In cases such as analysis of stators of traveling wave ultrasonic motors where only vibration modes with particular numbers of nodal diameters are of interest, the proposed approach is very convenient and useful.     

On reducible monodromy representations of some generalized Lamé equation

In this note, we compute the explicit formula of the monodromy data for a generalized Lamé equation when its monodromy is reducible but not completely reducible. We also solve the corresponding Riemann–Hilbert problem.

     

Simple zero property of some holomorphic functions on the moduli space of tori

We prove that some holomorphic functions on the moduli space of tori have only simple zeros.Instead of computing the derivative with respect to the moduli parameter τ, we introduce a conceptual proof by applying Painlevé Ⅵ equation. As an application of this simple zero property, we obtain the smoothness of the degeneracy curves of trivial critical points for some multiple Green function.