Graphs with small diameter determined by their DD-spectra

Let $G$ be a connected graph with vertex set $V(G)=\{v_{1},v_{2},...,v_{n}\}$. The distance matrix $D(G)=(d_{ij})_{n\times n}$ is the matrix indexed by the vertices of $G,$ where $d_{ij}$ denotes the distance between the vertices $v_{i}$ and $v_{j}$. Suppose that $\lambda_{1}(D)\geq\lambda_{2}(D)\geq\cdots\geq\lambda_{n}(D)$ are the distance spectrum of $G$. The graph $G$ is said to be determined by its $D$-spectrum if with respect to the distance matrix $D(G)$, any graph having the same spectrum as $G$ is isomorphic to $G$. In this paper, we give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs are determined by their $D$-spectra

Slalom in complex time: emergence of low-energy structures in tunnel ionization via complex time contours

The ionization of atoms by strong, low-frequency fields can generally be described well by assuming that the photoelectron is, after the ionization step, completely at the mercy of the laser field. However, certain phenomena, like the recent discovery of low-energy structures in the long-wavelength regime, require the inclusion of the Coulomb interaction with the ion once the electron is in the continuum. We explore the first-principles inclusion of this interaction, known as analytical R-matrix theory, and its consequences on the corresponding quantum orbits. We show that the trajectory must have an imaginary component, and that this causes branch cuts in the complex time plane when the real trajectory revisits the neighbourhood of the ionic core. We provide a framework for consistently navigating these branch cuts based on closest-approach times, which satisfy the equation $\mathbf{r}(t) \cdot \mathbf{v}(t) = 0$ in the complex plane. We explore the geometry of these roots and describe the geometrical structures underlying the emergence of LES in both the classical and quantum domains.Comment: Supplementary information at http://episanty.github.io/Slalom-in-complex-time

Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph

For a given graph G and an associated class of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in G, the collection of all possible spectra for such matrices is considered. Building on the pioneering work of Colin de Verdiere in connection with the Strong Arnold Property, two extensions are devised that target a better understanding of all possible spectra and their associated multiplicities. These new properties are referred to as the Strong Spectral Property and the Strong Multiplicity Property. Finally, these ideas are applied to the minimum number of distinct eigenvalues associated with G, denoted by q(G). The graphs for which q(G) is at least the number of vertices of G less one are characterized.Comment: 26 pages; corrected statement of Theorem 3.5 (a

Comparative Direct Analysis of Type Ia Supernova Spectra. IV. Postmaximum

A comparative study of optical spectra of Type Ia supernovae (SNe Ia) obtained near 1 week, 3 weeks, and 3 months after maximum light is presented. Most members of the four groups that were defined on the basis of maximum light spectra in Paper II (core normal, broad line, cool, and shallow silicon) develop highly homogeneous postmaximum spectra, although there are interesting exceptions. Comparisons with SYNOW synthetic spectra show that most of the spectral features can be accounted for in a plausible way. The fits show that 3 months after maximum light, when SN Ia spectra are often said to be in the nebular phase and to consist of forbidden emission lines, the spectra actually remain dominated by resonance scattering features of permitted lines, primarily those of Fe II. Even in SN 1991bg, which is said to have made a very early transition to the nebular phase, there is no need to appeal to forbidden lines at 3 weeks postmaximum, and at 3 months postmaximum the only clear identification of a forbidden line is [Ca II] 7291, 7324. Recent studies of SN Ia rates indicate that most of the SNe Ia that have ever occurred have been "prompt" SNe Ia, produced by young (100,000,000 yr) stellar populations, while most of the SNe Ia that occur at low redshift today are "tardy", produced by an older (several Gyrs) population. We suggest that the shallow silicon SNe Ia tend to be the prompt ones.Comment: Accepted by PAS