Let G = (V,E), V = {1,2, . . . ,n}, be a simple graph without isolated vertices, with vertex degree sequence d1 ≥ d2 ≥ ·· · ≥ dn > 0, di = d(i). A graph G is regular if and only if d1 = d2 = · · · = dn. A graph invariant I(G) is measure of irregularity of graph G with the property I(G) = 0 if and only ifG is regular, and I(G)> 0 otherwise. In this paper we introduce some new irregularity measures.

gC(μ) := { F0(C) Fμ(C) if Fμ(C) > 0 ; +∞ if Fμ(C) = 0. Then fC is an upper semicontinuous function: if μ ∈ SM is such that Fμ(C) > 0, then fC is continuous at point μ. Otherwise, fC(μ) = ∞ and fC is trivially upper semicontinuous at point μ. Clearly, one has G(μ) = infC∈C fC(μ) ; as an infimum of upper semicontinuous functions, it is itself upper semicontinuous, and therefore attains its maximu...

For two dimensional Schrödinger operator H with point interactions, we prove that wave operators of scattering for the pair $$(H,H_0)$$ , $$H_0$$ being free operator, are bounded in Lebesgue space $$L^p({{\mathbb {R}}}^2)$$ $$1<p<\infty $$ if and only there no generalized eigenfunctions $$Hu(x)=0$$ which satisfy $$u(x)= C|x|^{-1}+ o(|x|^{-1})$$ as $$|x|\rightarrow \infty $$C\not =0$$ . Otherwis...

THIS APPENDIX CONTAINS PROOFS of the results contained in the main paper. All notation is as defined in the main text unless explicitly noted otherwise. Equation numbering continues in sequence with that established in the main text. To simplify notation, let β denote the true parameter value β0 unless explicitly stated otherwise; similarly, the 0 subscript is removed from other objects, such a...