Showing papers by "Steve Kirkland published in 2010"
TL;DR: This paper considers the energy of a simple graph with respect to its normalized Laplacian eigenvalues, which is called the L-energy, and provides upper and lower bounds for L- energy based on its general Randic index R-1(G).
122 citations
TL;DR: In this article, it was shown that the Kemeny constant is bounded from above as T ranges over the irreducible members of a strongly connected directed graph if and only if D is an intercyclic directed graph.
39 citations
TL;DR: In this paper, it was shown that for any graph on n ≥ 5 vertices, 2 ≤ s(Q(G)) ≤ 2n − 4, and the equality cases in both bounds were characterized.
36 citations
TL;DR: A new formula for the Kemeny constant is presented and several perturbation results for the constant are developed, including conditions under which it is a convex function and for chains whose transition matrix has a certain directed graph structure.
Abstract: A quantity known as the Kemeny constant, which is used to measure the expected number of links that a surfer on the World Wide Web, located on a random web page, needs to follow before reaching his/her desired location, coincides with the more well known notion of the expected time to mixing, i.e., to reaching stationarity of an ergodic Markov chain. In this paper we present a new formula for the Kemeny constant and we develop several perturbation results for the constant, including conditions under which it is a convex function. Finally, for chains whose transition matrix has a certain directed graph structure we show that the Kemeny constant is dependent only on the common length of the cycles and the total number of vertices and not on the specific transition probabilities of the chain.
34 citations
TL;DR: The function @k yields a measure of vertex centrality, and the measure is applied to analyse certain graphs arising from food webs.
14 citations
TL;DR: For an irreducible stochastic matrix T, the sensitivity of the stationary distribution to perturbations in T was studied in this article, where column sum vectors for T provided information on T.
Abstract: For an irreducible stochastic matrix T, we consider a certain condition
number (T), which measures the sensitivity of the stationary distribution
vector to perturbations in T, and study the extent to which the column sum
vector for T provides information on (T). Specifically, if cT is the column
sum vector for some stochastic matrix of order n, we define the set S(c) =
{A|A is an n × n stochastic matrix with column sum vector cT }. We then
characterise those vectors cT such that (T) is bounded as T ranges over the
irreducible matrices in S(c); for those column sum vectors cT for which is
bounded, we give an upper bound on in terms of the entries in cT , and
characterise the equality case.
6 citations
TL;DR: In this paper, the maximum number of 0 s in an irreducible, totally nonnegative ( 0, 1 ) -matrix of order n is ( n - 1 ) 2 and the minimum Perron value of such matrices is 2 + 2 cos 2 π n + 2.
5 citations
01 Jan 2010
TL;DR: In this paper, the discretization of switched and non-switched linear positive systems using Pade approximations is considered, and sufficient conditions on the Pade approximation are given to preserve positivity of the discrete-time system.
Abstract: In this paper the discretization of switched and
non-switched linear positive systems using Pade approximations
is considered. We show:
1) first order diagonal Pade approximation preserves both
linear and quadratic co-positive Lyapunov functions,
higher order transformations need an additional condition
on the sampling time1;
2) positivity need not be preserved even for arbitrarily small
sampling time for certain Pade approximations.
Sufficient conditions on the Pade approximations are given to
preserve positivity of the discrete-time system. Finally, some
examples are given to illustrate the efficacy of our results.
2 citations