Showing papers by "Steve Kirkland published in 2009"
TL;DR: In this paper, the scrambling index of a primitive digraph D is defined as the smallest positive integer k such that for every pair of vertices u and v, there is a vertex w such that we can get to w from u and V in D by directed walks of length k; it is denoted by k(D).
50 citations
TL;DR: The scrambling index of a primitive digraph D is the smallest positive integer k such that for every pair of vertices u and v, there is a vertex w such that we can get to w from u and V in D by directed walks of length k ; it is denoted by k(D) as mentioned in this paper.
33 citations
TL;DR: In this article, a class of graphs whose adjacency matrices are nonsingular with integral inverses, denoted h-graphs, is presented, and necessary and sufficient conditions for the existence of G + are given.
24 citations
TL;DR: Given a primitive stochastic matrix, an upper bound on the moduli of its non-Perron eigenvalues is provided in terms of the weights of the cycles in the directed graph associated with the matrix.
Abstract: Given a primitive stochastic matrix, we provide an upper bound on the moduli of its non-Perron eigenvalues. The bound is given in terms of the weights of the cycles in the directed graph associated with the matrix. The bound is attainable in general, and we characterize a special case of equality when the stochastic matrix has a positive row. Applications to Leslie matrices and to Google-type matrices are also considered.
13 citations
TL;DR: This paper provides a constructive characterization of the graphs, G, for which $\Lambda(G)$ and $D^*( G)$ share all but two elements.
Abstract: A conjecture of Grone and Merris states that for any graph $G$, its Laplacian spectrum, $\Lambda(G)$, is majorized by its conjugate degree sequence, $D^*(G)$. That conjecture prompts an investigation of the relationship between $\Lambda(G)$ and $D^*(G),$ and Merris has characterized the graphs $G$ for which the multisets $\Lambda(G)$ and $D^*(G)$ are equal. In this paper, we provide a constructive characterization of the graphs $G$ for which $\Lambda(G)$ and $D^*(G)$ share all but two elements.
3 citations
TL;DR: In this paper, the authors give necessary and sufficient conditions for the occurrence of Q-spectral integral variation only in two places, as the first case never occurs, while the second case always occurs.
2 citations