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Constructions for type I trees with nonisomorphic Perron branches

Stephen J. Kirkland — 1999

Czechoslovak Mathematical Journal

A tree is classified as being type I provided that there are two or more Perron branches at its characteristic vertex. The question arises as to how one might construct such a tree in which the Perron branches at the characteristic vertex are not isomorphic. Motivated by an example of Grone and Merris, we produce a large class of such trees, and show how to construct others from them. We also investigate some of the properties of a subclass of these trees. Throughout, we exploit connections between...

Random walk centrality and a partition of Kemeny's constant

Stephen J. Kirkland — 2016

Czechoslovak Mathematical Journal

We consider an accessibility index for the states of a discrete-time, ergodic, homogeneous Markov chain on a finite state space; this index is naturally associated with the random walk centrality introduced by Noh and Reiger (2004) for a random walk on a connected graph. We observe that the vector of accessibility indices provides a partition of Kemeny's constant for the Markov chain. We provide three characterizations of this accessibility index: one in terms of the first return time to the state...

On a bound on algebraic connectivity: the case of equality

Stephen J. KirklandNeumann, MichaelBryan L. Shader — 1998

Czechoslovak Mathematical Journal

In a recent paper the authors proposed a lower bound on 1 - λ i , where λ i , λ i 1 , is an eigenvalue of a transition matrix T of an ergodic Markov chain. The bound, which involved the group inverse of I - T , was derived from a more general bound, due to Bauer, Deutsch, and Stoer, on the eigenvalues of a stochastic matrix other than its constant row sum. Here we adapt the bound to give a lower bound on the algebraic connectivity of an undirected graph, but principally consider the case of equality in the bound when...

Bounds on the subdominant eigenvalue involving group inverses with applications to graphs

Stephen J. KirklandNeumann, MichaelBryan L. Shader — 1998

Czechoslovak Mathematical Journal

Let A be an n × n symmetric, irreducible, and nonnegative matrix whose eigenvalues are λ 1 > λ 2 ... λ n . In this paper we derive several lower and upper bounds, in particular on λ 2 and λ n , but also, indirectly, on μ = max 2 i n | λ i | . The bounds are in terms of the diagonal entries of the group generalized inverse, Q # , of the singular and irreducible M-matrix Q = λ 1 I - A . Our starting point is a spectral resolution for Q # . We consider the case of equality in some of these inequalities and we apply our results to the algebraic connectivity of undirected...

Algebraic connectivity of k -connected graphs

Stephen J. KirklandIsrael RochaVilmar Trevisan — 2015

Czechoslovak Mathematical Journal

Let G be a k -connected graph with k 2 . A hinge is a subset of k vertices whose deletion from G yields a disconnected graph. We consider the algebraic connectivity and Fiedler vectors of such graphs, paying special attention to the signs of the entries in Fiedler vectors corresponding to vertices in a hinge, and to vertices in the connected components at a hinge. The results extend those in Fiedler’s papers Algebraic connectivity of graphs (1973), A property of eigenvectors of nonnegative symmetric...

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