Showing papers by "Steve Kirkland published in 1997"
TL;DR: This work investigates 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, and produces a lower bound on the algebraIC connectivity of any unweighted graph in terms of the diameter and the number of vertices.
Abstract: 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.
58 citations
TL;DR: In this paper, for a primitive generalized tournament matrix, upper and lower bounds on an entry in its perron vector in terms of the corresponding row sum of the matrix are presented.
14 citations
TL;DR: In this paper, the general form of the characteristic polynomial of a primitive stochastic matrix of order n ⩾ 7 and exponent at least 2⌊ (n − 4) 4 ⌋ complex eigenvalues of modulus at greater than { 1 2 sin [ π(n − 1) ]} 2 (n−1) ]−2 (n-1) (observed that this last quantity tends to 1 as n → ∞).
11 citations
TL;DR: In this article, it was shown that the ranking schemes of Kendall and Wei and of Ramanujacharyula agree with the ranking generated by the row sums of the tournament matrix T.
9 citations
TL;DR: For a primitive matrix A of order n + k having a primitive submatrix of order 71, the exponent of A is at most (n - 1) + 2k + 1.
1 citations