Stephen J. Kirkland

Distances in Weighted Trees and Group Inverse of Laplacian Matrices

In this paper we find formulas for group inverses of Laplacians of weighted trees. We then develop a relationship between entries of the group inverse and various distance functions on trees. In particular, we show that the maximal and minimal entries on the diagonal of the group inverse correspond to certain pendant vertices of the tree and to a centroid of the tree, respectively. We also give a characterization for the group inverses of the Laplacian of an unweighted tree to be an M-matrix.

The spread of the spectrum of a graph

Upper and lower bounds are obtained for the spread λ1 − λn of the eigenvalues λ1 λ2 ··· λn of the adjacency matrix of a simple graph.

A Google-like model of road network dynamics and its application to regulation and control

Inspired by the ability of Markov chains to model complex dynamics and handle large volumes of data in Googles PageRank algorithm, a similar approach is proposed here to model road network dynamics. The central component of the Markov chain is the transition matrix which can be completely constructed by easily collecting traffic data. The proposed model is validated using the popular mobility simulator SUMO. Markov chain theory and spectral analysis of the transition matrix are then shown to reveal non-evident properties of the underlying road network and to correctly predict consequences of road network modifications. Preliminary results from possible applications are shown and simple practical examples are provided throughout this article to clarify and support the theoretical expectations.

On graphs with equal algebraic and vertex connectivity

Let G be an undirected unweighted graph on n vertices, let L be its Laplacian matrix, and let L#=(l#i,j) be the group inverse of L. It is known that for Z(L#):=(1/2)max1⩽i,j⩽n∑s=1n|li,s#−lj,s#|, the quantity 1/Z(L#) is a lower bound on the algebraic connectivity a(G) of G, while the vertex connectivity of G, v(G), is an upper bound on a(G). We characterize the graphs G for which v(G)=a(G) and subsequently prove that if n⩾v(G)2, then v(G)=a(G) holds if and only if 1/Z(L#)=a(G)=v(G). We close with an example showing that the equality 1/Z(L#)=a(G) does not necessarily imply that 1/Z(L#)=a(G)=v(G).

Aztec diamonds and digraphs, and Hankel determinants of Schröder numbers

The Aztec diamond of order n is a certain configuration of 2n(n + 1) unit squares. We give a new proof of the fact that the number Πn of tilings of the Aztec diamond of order n with dominoes equals 2n(n+1)/2. We determine a sign-nonsingular matrix of order n (n + 1) whose determinant gives Πn. We reduce the calculation of this determinant to that of a Hankel matrix of order n whose entries are large Schroder numbers. To calculate that determinant we make use of the J-fraction expansion of the generating function of the Schroder numbers.

APPLICATIONS OF PAZ'S INEQUALITY TO PERTURBATION BOUNDS FOR MARKOV CHAINS

Abstract In several papers Meyer, singly and with coauthors, established the usefulness of the group generalized inverse in the study and computations of various aspects of Markov chains. Here we are interested in those results which concern bounds on the condition number of the chain and on the error in the computation of the stationary distribution vector. We show that a lemma due to Paz can be used to improve, sometimes by a factor of 2, some of the constants in the bounds obtained in the aforementioned papers.

On a Question Concerning Condition Numbers for Markov Chains

Let S be an irreducible stochastic matrix of order n with left stationary vector

Tournament matrices with extremal spectral properties

\pi^T,

On a bound on algebraic connectivity: the case of equality

and let S(i) denote the principal submatrix of S formed by deleting the ith row and column. We prove that

Convexity and Concavity of the Perron Root and Vector of LeslieMatrices with Applications to a Population Model

\max_{1 \le i \le n}\pi_i||(I- S_{(i)})^{-1}||_{\infty} \le \min_{1 \le j \le n}||(I- S_{(j)})^{-1}||_{\infty},