The analog of the principal SO(3) subalgebra of a finite dimensional simple Lie algebra can be defined for any hyperbolic Kac Moody algebra g(A) associated with a symmetrizable Cartan matrix A, and coincides with the non-compact group SO(1, 2). We exhibit the decomposition of g(A) into representations of SO(1, 2); with the exception of the adjoint SO(1, 2) algebra itself, all of these represent...

A graph is called of type k if it is connected, regular, and has k distinct eigenvalues. For example graphs of type 2 are the complete graphs, while those of type 3 are the strongly regular graphs. We prove that for any positive integer n, every graph can be embedded in n cospectral, non-isomorphic graphs of type k for every k ≥ 3. Furthermore, in the case k ≥ 5 such a family of extensions can ...

Assessing network systems for failures is critical to mitigate the risk and develop proactive responses. In this paper, we investigate devastating consequences of link failures in networks. We propose an exact algorithm and a spectral lowerbound on the minimum number of removed links to incur a significant level of disruption. Our exact solution can identify optimal solutions in both uniform an...

We study the sum ζH(s) = ∑ j E j over the eigenvalues Ej of the Schrˇdinger equation in a spherical domain with Dirichlet walls, threaded by a line of magnetic flux. Rather than using Green’s function techniques, we tackle the mathematically nontrivial problem of finding exact sum rules for the zeros of Bessel functions Jν , which are extremely helpful when seeking numerical approximations to g...

The spectral problem of the three-dimensional manifolds M3 A admitting Sol-geometry in Thurston’s sense is investigated. Topologically M3 A are the torus bundles over a circle with a hyperbolic glueing map A. The eigenfunctions of the corresponding Laplace-Beltrami operators are described in terms of the modified Mathieu functions. It is shown that the multiplicities of the eigenvalues do not d...

The Clement or Sylvester-Kac matrix is a tridiagonal matrix with zero diagonal and simple integer entries. Its spectrum is known explicitly and consists of integers which makes it a useful test matrix for numerical eigenvalue computations. We consider a new class of appealing two-parameter extensions of this matrix which have the same simple structure and whose eigenvalues are also given explic...

We prove an integrability criterion of order 3 for a homogeneous potential of degree−1 in the plane. Still, this criterion depends on some integer and it is impossible to apply it directly except for families of potentials whose eigenvalues are bounded. To address this issue, we use holonomic and asymptotic computations with error control of this criterion and apply it to the potential of the f...

For an arbitrary simple graph G and a positive integer r, the super line multigraph of index r of G, denoted Mr(G), has for vertices all the r-subsets of edges. Two vertices S and T are joined by as many edges as pairs of distinct edges s ∈ S and t ∈ T share a common vertex in G. We present spectral properties ofMr(G) and particularly, if G is a regular graph, we calculate all the eigenvalues o...

Deo and Micikevicius recently gave a new bijection for spanning trees of complete bipartite graphs. In this paper we devise a generalization of Deo and Micikevicius’s method, which is also a modification of Olah’s method for encoding the spanning trees of any complete multipartite graph K(n1, . . . , nr). We also give a bijection between the spanning trees of a planar graph and those of any of ...

We show that termination of a simple class of linear loops over the integers is decidable. Namely we show that termination of deterministic linear loops is decidable over the integers in the homogeneous case, and over the rationals in the general case. This is done by analyzing the powers of a matrix symbolically using its eigenvalues. Our results generalize the work of Tiwari [Tiw04], where si...