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

Stephen J. Kirkland; Neumann, Michael; Bryan L. Shader

Czechoslovak Mathematical Journal (1998)

- Volume: 48, Issue: 1, page 1-20
- ISSN: 0011-4642

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topKirkland, 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 -

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