39 research outputs found
Clustering of spectra and fractals of regular graphs
We exhibit a characteristic structure of the class of all regular graphs of
degree d that stems from the spectra of their adjacency matrices. The structure
has a fractal threadlike appearance. Points with coordinates given by the mean
and variance of the exponentials of graph eigenvalues cluster around a line
segment that we call a filar. Zooming-in reveals that this cluster splits into
smaller segments (filars) labeled by the number of triangles in graphs. Further
zooming-in shows that the smaller filars split into subfilars labelled by the
number of quadrangles in graphs, etc. We call this fractal structure,
discovered in a numerical experiment, a multifilar structure. We also provide a
mathematical explanation of this phenomenon based on the Ihara-Selberg trace
formula, and compute the coordinates and slopes of all filars in terms of
Bessel functions of the first kind.Comment: 10 pages, 5 figure
Multivariate polynomial perturbations of algebraic equations
AbstractIn this note we study multivariate perturbations of algebraic equations. In general, it is not possible to represent the perturbed solution as a Puiseux-type power series in a connected neighborhood. For the case of two perturbation parameters we provide a sufficient condition that guarantees such a representation. Then, we extend this result to the case of more than two perturbation parameters. We motivate our study by the perturbation analysis of a weighted random walk on the Web Graph. In an instance of the latter the stationary distribution of the weighted random walk, the so-called Weighted PageRank, may depend on two (or more) perturbation parameters in a manner that illustrates our theoretical development
Consistent behavior of certain perturbed determinants induced by graphs
Abstract not availableVladimir Ejov, Giang T. Nguye