On group inverses of M-matrices with uniform diagonal entries
✍ Scribed by Stephen J. Kirkland; Michael Neumann
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 146 KB
- Volume
- 296
- Category
- Article
- ISSN
- 0024-3795
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✦ Synopsis
In two recent works the condition of the diagonal entries of the group inverse of a singular and irreducible M-matrix being uniform (constant) has arisen: in resistive electrical circuits and in the eect upon the Perron root of certain diagonal perturbation of a nonnegative matrix. In this paper we ®rst show a negative result that the group inverse of the Laplacian matrix of an undirected weighted graph G on n vertices with a cutpoint cannot have uniform diagonal. This includes the case when G is a tree. We characterize, however, all weighted n-cycles G whose Laplacian has a group inverse with a uniform diagonal. Finally, we consider the mean ®rst passage matrix M of an ergodic Markov chain with a doubly stochastic transition matrix T. We show that if the group inverse of s À has a uniform diagonal, then the group inverse of the M-matrix qws À w is again an M-matrix.