1,512 research outputs found
Merging the A- and Q-spectral theories
Let be a graph with adjacency matrix , and let
be the diagonal matrix of the degrees of The signless
Laplacian of is defined as .
Cvetkovi\'{c} called the study of the adjacency matrix the %
\textit{-spectral theory}, and the study of the signless Laplacian--the
\textit{-spectral theory}. During the years many similarities and
differences between these two theories have been established. To track the
gradual change of into in this paper it
is suggested to study the convex linear combinations of and defined by This study sheds new light
on and , and yields some surprises, in
particular, a novel spectral Tur\'{a}n theorem. A number of challenging open
problems are discussed.Comment: 26 page
Spectral properties of the hierarchical product of graphs
The hierarchical product of two graphs represents a natural way to build a
larger graph out of two smaller graphs with less regular and therefore more
heterogeneous structure than the Cartesian product. Here we study the
eigenvalue spectrum of the adjacency matrix of the hierarchical product of two
graphs. Introducing a coupling parameter describing the relative contribution
of each of the two smaller graphs, we perform an asymptotic analysis for the
full spectrum of eigenvalues of the adjacency matrix of the hierarchical
product. Specifically, we derive the exact limit points for each eigenvalue in
the limits of small and large coupling, as well as the leading-order relaxation
to these values in terms of the eigenvalues and eigenvectors of the two smaller
graphs. Given its central roll in the structural and dynamical properties of
networks, we study in detail the Perron-Frobenius, or largest, eigenvalue.
Finally, as an example application we use our theory to predict the epidemic
threshold of the Susceptible-Infected-Susceptible model on a hierarchical
product of two graphs
Trace formulae for Schrodinger operators on metric graphs with applications to recovering matching conditions
The paper is a continuation of the study started in \cite{Yorzh1}.
Schrodinger operators on finite compact metric graphs are considered under the
assumption that the matching conditions at the graph vertices are of
type. Either an infinite series of trace formulae (provided that edge
potentials are infinitely smooth) or a finite number of such formulae (in the
cases of and edge potentials) are obtained which link together two
different quantum graphs under the assumption that their spectra coincide.
Applications are given to the problem of recovering matching conditions for a
quantum graph based on its spectrum.Comment: arXiv admin note: substantial text overlap with arXiv:1403.761
Photoelectron spectra in an autoionization system interacting with a neighboring atom
Photoelectron ionization spectra of an autoionization system with one
discrete level interacting with a neighbor two-level atom are discussed. The
formula for long-time ionization spectra is derived. According to this formula,
the spectra can be composed of up to eight peaks. Moreover, the Fano-like zeros
for weak optical pumping have been identified in these spectra. The conditional
ionization spectra depending on the state of the neighbor atom exhibit
oscillations at the Rabi frequency. Dynamical spectral zeros occurring once per
the Rabi period have been revealed in these spectra.Comment: 10 pages, 13 figure
Essential spectra of difference operators on \sZ^n-periodic graphs
Let (\cX, \rho) be a discrete metric space. We suppose that the group
\sZ^n acts freely on and that the number of orbits of with respect to
this action is finite. Then we call a \sZ^n-periodic discrete metric
space. We examine the Fredholm property and essential spectra of band-dominated
operators on where is a \sZ^n-periodic discrete metric space.
Our approach is based on the theory of band-dominated operators on \sZ^n and
their limit operators.
In case is the set of vertices of a combinatorial graph, the graph
structure defines a Schr\"{o}dinger operator on in a natural way. We
illustrate our approach by determining the essential spectra of Schr\"{o}dinger
operators with slowly oscillating potential both on zig-zag and on hexagonal
graphs, the latter being related to nano-structures
- …