‎let $g$ be a graph without an isolated vertex‎, ‎the normalized laplacian matrix $tilde{mathcal{l}}(g)$‎‎is defined as $tilde{mathcal{l}}(g)=mathcal{d}^{-frac{1}{2}}mathcal{l}(g) mathcal{d}^{-frac{1}{2}}$‎, where ‎$‎mathcal{‎d}‎$ ‎is a‎ diagonal matrix whose entries are degree of ‎vertices ‎‎of ‎$‎g‎$‎‎. ‎the eigenvalues of‎‎$tilde{mathcal{l}}(g)$ are ‎called ‎ ‎ as ‎the ‎normalized laplacian ...

In this paper, we consider localized integral operators whose kernels have mild singularity near the diagonal and certain Hölder regularity and decay off the diagonal. Our model example is the Bessel potential operator Jγ , γ > 0. We show that if such a localized integral operator has stability on a weighted function space Lw for some p ∈ [1,∞) and Muckenhoupt Ap-weight w, then it has stability...

We prove that if (a; b) is an R-diagonal pair in some non-commutative probability space (A; ') then (a p ; b p) is R-diagonal too and we compute the determining series f (a p ;b p) in terms of the distribution of ab. We give estimates of the upper and lower bounds of the support of free multiplicative convolution of probability measures compactly supported on 0; 1, and use the results to give n...