41 research outputs found
Interlace Polynomials of Friendship Graphs
In this paper, we study the interlace polynomials of friendship graphs, that is, graphs that satisfy the Friendship Theorem given by Erdös, Rényi and Sos. Explicit formulas, special values and behavior of coefficients of these polynomials are provided. We also give the interlace polynomials of other similar graphs, such as, the butterfly graph
Graphs with few matching roots
We determine all graphs whose matching polynomials have at most five distinct
zeros. As a consequence, we find new families of graphs which are determined by
their matching polynomial.Comment: 14 pages, 7 figures, 1 appendix table. Final version. Some typos are
fixe
Interlace Polynomials of Certain Graphs
In this research, we investigated the interlace polynomials of a shell graph as well as other related graphs. A shell graph, Tn is constructed by adding edges to a cycle graph such that all vertices are adjacent to one vertex. The main results of this thesis include iterative and explicit formulas for the interlace polynomial of a shell graph, denoted q(Tn; x). A linear algebra application using the adjacency matrices of the chosen graphs is also explored
Spectra of Quantum Trees and Orthogonal Polynomials
We investigate the spectrum of regular quantum-graph trees, where the edges are endowed with a Schr\ odinger operator with self-adjoint Robin vertex conditions. It is known that, for large eigenvalues, the Robin spectrum approaches the Neumann spectrum. In this research, we compute the lower Robin spectrum. The spectrum can be obtained from the roots of a sequence of orthogonal polynomials involving two variables. As the length of the quantum tree increases, the spectrum approaches a band-gap structure. We find that the lowest band tends to minus infinity as the Robin parameter increases, whereas the rest of the bands remain positive. Unexpectedly, we find that two groups of isolated negative eigenvalues separate from the bottom of the lowest band. These eigenvalues are computed as they depend asymptotically on the Robin parameter. Our analysis invokes the interlacing property of orthogonal polynomials
The Spectrum of Random Inner-product Kernel Matrices
We consider n-by-n matrices whose (i, j)-th entry is f(X_i^T X_j), where X_1,
...,X_n are i.i.d. standard Gaussian random vectors in R^p, and f is a
real-valued function. The eigenvalue distribution of these random kernel
matrices is studied at the "large p, large n" regime. It is shown that, when p
and n go to infinity, p/n = \gamma which is a constant, and f is properly
scaled so that Var(f(X_i^T X_j)) is O(p^{-1}), the spectral density converges
weakly to a limiting density on R. The limiting density is dictated by a cubic
equation involving its Stieltjes transform. While for smooth kernel functions
the limiting spectral density has been previously shown to be the
Marcenko-Pastur distribution, our analysis is applicable to non-smooth kernel
functions, resulting in a new family of limiting densities
Graphs with Few Eigenvalues. An Interplay between Combinatorics and Algebra.
Abstract: Two standard matrix representations of a graph are the adjacency matrix and the Laplace matrix. The eigenvalues of these matrices are interesting parameters of the graph. Graphs with few eigenvalues in general have nice combinatorial properties and a rich structure. A well investigated family of such graphs comprises the strongly regular graphs (the regular graphs with three eigenvalues), and we may see other graphs with few eigenvalues as algebraic generalizations of such graphs. We study the (nonregular) graphs with three adjacency eigenvalues, graphs with three Laplace eigenvalues, and regular graphs with four eigenvalues. The last ones are also studied in relation with three-class association schemes. We also derive bounds on the diameter and on the size of special subsets in terms of the eigenvalues of the graph. Included are lists of feasible parameter sets of graphs with three Laplace eigenvalues, regular graphs with four eigenvalues, and three-class association schemes.