Steve Kirkland

Extremizing algebraic connectivity subject to graph theoretic constraints

The main problem of interest is to investigate how the algebraic connectivity o f a weighted connected graph behaves when the graph is perturbed by removing one or more connected components at a xed vertex and replacing this collection by a single connected component. This analysis leads to exhibiting the unique up to isomorphismtrees on n vertices with speciied diameter that maximize and minimize the algebraic connectivity o ver all such trees. When the radius of a graph is the speciied constraint the unique minimizer of the algebraic connectivity o ver all such graphs is also determined. Analogous results are proved for unicyclic graphs with xed girth. In particular, the unique minimizer and maximizer of the algebraic connectivity is given over all such graphs with girth 3.

Characteristic vertices of weighted trees via perron values

We consider a weighted tree T with algebraic connectivity μ, and characteristic vertex v. We show that μ and its associated eigenvectors can be described in terms of the Perron value and vector of a nonnegative matrix which can be computed from the branches of T at v. The machinery of Perron-Frobenius theory can then be used to characterize Type I and Type II trees in terms of these Perron values, and to show that if we construct a weighted tree by taking two weighted trees and identifying a vertex of one with a vertex of the other, then any characteristic vertex of the new tree lies on the path joining the characteristic vertices of the two old trees.

Convergence Analysis of a PageRank Updating Algorithm by Langville and Meyer

The PageRank updating algorithm proposed by Langville and Meyer is a special case of an iterative aggregation/disaggregation (SIAD) method for computing stationary distributions of very large Markov chains. It is designed, in particular, to speed up the determination of PageRank, which is used by the search engine Google in the ranking of web pages. In this paper the convergence, in exact arithmetic, of the SIAD method is analyzed. The SIAD method is expressed as the power method preconditioned by a partial LU factorization. This leads to a simple derivation of the asymptotic convergence rate of the SIAD method. It is known that the power method applied to the Google matrix always converges, and we show that the asymptotic convergence rate of the SIAD method is at least as good as that of the power method. Furthermore, by exploiting the hyperlink structure of the web it can be shown that the asymptotic convergence rate of the SIAD method applied to the Google matrix can be made strictly faster than that of the power method.

Algebraic connectivity of weighted trees under perturbation

We investigate how the algebraic connectivity of a weighted tree behaves when the tree is perturbed by removing one of its branches and replacing it with another. This leads to a number of results, for example the facts that replacing a branch in an unweighted tree by a star on the same number of vertices will not decrease the algebraic connectivity, while replacing a certain branch by a path on the same number of vertices will not increase the algebraic connectivity. We also discuss how the arrangement of the weights on the edges of a tree affects the algebraic connectivity, and we produce a lower bound on the algebraic connectivity of any unweighted graph in terms of the diameter and the number of vertices. Throughout, our techniques exploit a connection between the algebraic connectivity of a weighted tree and certain positive matrices associated with the tree.

Perron components and algebraic connectivity for weighted graphs

The algebraic connectivity of a connected graph is the second-smallest eigenvalue of its Laplacian matrix, and a remarkable result of Fiedler gives information on the structure of the eigenvectors associated with that eigenvalue. In this paper, we introduce the notion of a perron component at a vertex in a weighted graph, and show how the structure of the eigenvectors associated with the algebraic connectivity can be understood in terms of perron components. This leads to some strengthening of Fiedlers original result, gives some insights into weighted graphs under perturbation, and allows for a discussion of weighted graphs exhibiting tree-like structure.

Algebraic multiplicity of the eigenvalues of a tournament matrix

Let F” denote the set of irreducible n X n tournament matrices. Here arc our main results: (1) For all n > 3, every matrix in K has at least three distinct eigenvalues; such a matrix has exactly three distinct eigenvalues if and only if it is a Hadamard tournament matrix. (2) For all n 2 3 there is a matrix in Y” having n distinct eigenvalues. (3) If cr, denotes the maximum algebraic multiplicity of 0 as an eigenvalue of the matrices in z, then ln /21-2 6. (4) If r,, is the minimum Perron value (i.e. spectral radius) of all matrices in r”, then 2 8.

A bound on the algebraic connectivity of a graph in terms of the number of cutpoints

Let G be a graph on n vertices which has k cutpoints. For the case that 2≤k≤n/2, we prove tha the algebraic connectivity of G is at most , and we explicitly characterize the graphs attaining equality in this bound.

Minimizing algebraic connectivity over connected graphs with fixed girth

Abstract Let G n , g denote the class of all connected graphs on n vertices with fixed girth g . We prove that if n ⩾3 g −1, then the graph which uniquely minimizes the algebraic connectivity over G n , g is the unicyclic “lollipop” graph C n , g obtained by appending a g cycle to a pendant vertex of a path on n − g vertices. The characteristic set of C n , g is also discussed. Throughout both algebraic and combinatorial techniques are used.

Hypertournament matrices, score vectors and eigenvalues

Consider a real square matrix A of order n which satisfies A+ At = J− I (where J is the all ones matrix) and its score vectors= Al (where 1 is the all ones vector). Here are our main results. If , then A has a real positive eigenvalue p with while the other eigenvalues satisfy A has eigenvalues p and λ such that and if and only if A has n − 2 eigenvalues with real part -1/2. If then for any eigenvalue . Further, if then a Perron-Frobenius result holds for A. A consequence of this is that if such an A is non-negative as well, and if n ≥ 9, then A is irreducible and primitive.

Pick's inequality and tournaments

Abstract Let T be a tournament on n nodes, and let A be its (adjacency) matrix. A. Brauer and I. Gentry observed that an inequality due to G. Pick implies that | Im λ|⩽ 1 2 cot ( π 2n ) for all eigenvalues λ of A. We say that T is a Pick (or P -) tournament if equality holds for some λ. We determine when equality holds in Picks inequality for arbitrary real matrices and use this to show that the P -tournaments can all be constructed from the transitive tournament M by reversing the arcs between the sets of certain node partitions or cuts {U, Ū}. The cuts are specified by ±1 n-vectors u such that utw=0, where w =[ 1 , σ,…, σ n− 1 ] t , σ=e iπ n . This links the cuts to cyclotomic polynomials. There is at least one P -tournament on n nodes if and only if n≠2k, k⩾1. Up to isomorphism, there is precisely one P -tournament on n nodes if (and only if) n=p or n=2p for some odd prime p. For odd n, up toisomorphism, the only regular P -tournament on n nodes has matrix Zn = Circ(0, 1,…, 1, 0,…, 0). A composition rule is used on the matrices Zp to form all P -tournaments on n=2kpl nodes, l⩾1.