ISSN:
1572-9141
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Let A be an n × n symmetric, irreducible, and nonnegative matrix whose eigenvalues are λ1〉λ2≥ ... ≥λn. In this paper we derive several lower and upper bounds, in particular on λ2 and λ n , but also, indirectly, on $$\mu = \mathop {\max }\limits_{2 \leqslant i \leqslant n} |\lambda _i |$$ . The bounds are in terms of the diagonal entries of the group generalized inverse, Q #, of the singular and irreducible M-matrix Q = λ1 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 #.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1022455208972
