Let & be a ¿-regular graph and T the covering tree of S. We define a cogrowth constant of & in T and express it in terms of the first eigenvalue of the Laplacian on S . As a corollary, we show that the cogrowth constant is as large as possible if and only if the first eigenvalue of the Laplacian on 9 is zero. Grigorchuk's criterion for amenability of finitely generated groups follows. In this n...

We analyze the Ritz–Galerkin method for symmetric eigenvalue problems and prove a priori eigenvalue error estimates. For a simple eigenvalue, we prove an error estimate that depends mainly on the approximability of the corresponding eigenfunction and provide explicit values for all constants. For a multiple eigenvalue we prove, in addition, apparently the first truly a priori error estimates th...

It is proved that applying sufficient regularity conditions to the interval matrix $[A-|B|,A + |B|]$, we can create a new unique solvability condition for the absolute value equation $Ax + B|x|=b$, since regularity of interval matrices implies unique solvability of their corresponding absolute value equation. This condition is formulated in terms of positive deniteness of a certain point matrix...

For a graph, the least signless Laplacian eigenvalue is the least eigenvalue of its signless Laplacian matrix. This paper investigates how the least signless Laplacian eigenvalue of a graph changes under some perturbations, and minimizes the least signless Laplacian eigenvalue among all the nonbipartite graphs with given matching number and edge cover number, respectively.

If A is an n × n matrix whose n eigenvalues are ordered in terms of decreasing modules, |λ1| ≥ |λ2| ≥ · · · |λn|, it is often of interest to estimate |λ2| |λ1| . If A is a row stochastic matrix (so λ1 = 1), one can use an old formula of R. L. Dobrushin to give a useful, explicit formula for |λ2|. The purpose of this note is to disseminate these known results more widely and to show how they imp...

In this article we characterize the L mathvariant="normal">∞ <mml:annotation encoding="application/...