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

Kirkland, Stephen J., Neumann, Michael, and Shader, Bryan L.. "Bounds on the subdominant eigenvalue involving group inverses with applications to graphs." Czechoslovak Mathematical Journal 48.1 (1998): 1-20. <http://eudml.org/doc/30397>.

@article{Kirkland1998,
abstract = {Let $A$ be an $n\times n$ symmetric, irreducible, and nonnegative matrix whose eigenvalues are $\lambda _1 > \lambda _2 \ge \ldots \ge \lambda _n$. In this paper we derive several lower and upper bounds, in particular on $\lambda _2$ and $\lambda _n$, but also, indirectly, on $\mu = \max _\{2\le i \le n\} |\lambda _i|$. The bounds are in terms of the diagonal entries of the group generalized inverse, $Q^\{\#\}$, of the singular and irreducible M-matrix $Q=\lambda _1 I-A$. 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 graphs, where now $Q$ becomes $L$, the Laplacian of the graph. In case the graph is a tree we find a graph-theoretic interpretation for the entries of $L^\{\#\}$ and we also sharpen an upper bound on the algebraic connectivity of a tree, which is due to Fiedler and which involves only the diagonal entries of $L$, by exploiting the diagonal entries of $L^\{\#\}$.},
author = {Kirkland, Stephen J., Neumann, Michael, Shader, Bryan L.},
journal = {Czechoslovak Mathematical Journal},
keywords = {nonnegative matrix; eigenvalue; lower and upper bounds; tree; undirected graph; group generalized inverse; irreducible -matrix; algebraic connectivity},
language = {eng},
number = {1},
pages = {1-20},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Bounds on the subdominant eigenvalue involving group inverses with applications to graphs},
url = {http://eudml.org/doc/30397},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Kirkland, Stephen J.
AU - Neumann, Michael
AU - Shader, Bryan L.
TI - Bounds on the subdominant eigenvalue involving group inverses with applications to graphs
JO - Czechoslovak Mathematical Journal
PY - 1998
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 48
IS - 1
SP - 1
EP - 20
AB - Let $A$ be an $n\times n$ symmetric, irreducible, and nonnegative matrix whose eigenvalues are $\lambda _1 > \lambda _2 \ge \ldots \ge \lambda _n$. In this paper we derive several lower and upper bounds, in particular on $\lambda _2$ and $\lambda _n$, but also, indirectly, on $\mu = \max _{2\le i \le n} |\lambda _i|$. The bounds are in terms of the diagonal entries of the group generalized inverse, $Q^{\#}$, of the singular and irreducible M-matrix $Q=\lambda _1 I-A$. 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 graphs, where now $Q$ becomes $L$, the Laplacian of the graph. In case the graph is a tree we find a graph-theoretic interpretation for the entries of $L^{\#}$ and we also sharpen an upper bound on the algebraic connectivity of a tree, which is due to Fiedler and which involves only the diagonal entries of $L$, by exploiting the diagonal entries of $L^{\#}$.
LA - eng
KW - nonnegative matrix; eigenvalue; lower and upper bounds; tree; undirected graph; group generalized inverse; irreducible -matrix; algebraic connectivity
UR - http://eudml.org/doc/30397
ER -