Bounds for eigenvalues of certain stochastic matrices

H. J. Landau, A. M. Odlyzko

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We consider the class of stochastic matrices M generated in the following way from graphs: if G is an undirected connected graph on n vertices with adjacency matrix A, we form M from A by dividing the entries in each row of A by their row sum. Being stochastic, M has the eigenvalue λ=1 and possibly also an eigenvalue λ=-1. We prove that the remaining eigenvalues of M lie in the disk |λ|≤1-n-3, and show by examples that the order of magnitude of this estimate is best possible. In these examples, G has a bar-bell structure, in which n/3 of the vertices are arranged along a line, with n/3 vertices fully interconnected at each end. We also obtain better bounds when either the diameter of G or the maximal degree of a vertex is restricted.

Original languageEnglish (US)
Pages (from-to)5-15
Number of pages11
JournalLinear Algebra and Its Applications
Issue numberC
StatePublished - Jun 1981
Externally publishedYes


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