TY - JOUR

T1 - Bounds for eigenvalues of certain stochastic matrices

AU - Landau, H. J.

AU - Odlyzko, A. M.

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 1981/6

Y1 - 1981/6

N2 - 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.

AB - 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.

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U2 - 10.1016/0024-3795(81)90003-3

DO - 10.1016/0024-3795(81)90003-3

M3 - Article

AN - SCOPUS:0039942204

SN - 0024-3795

VL - 38

SP - 5

EP - 15

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - C

ER -