Titlebook: ;

影響深遠(yuǎn)

ambivalence

Talkative

RecurrenceThe topic presented in this chapter is recurrence. This concept can be studied via probability, potential theory and operator theory and has interpretations in each context.

dermatomyositis

歡樂東方

Uniformly Positive MeasureIn this chapter we look at consequences of lower bounds on the measure . for a graph . over a discrete measure space .. We formulate the lower bound assumptions in two ways.

Armory

格言

Sparseness and Isoperimetric InequalitiesIn this chapter we investigate what it means for a graph to have relatively few edges. This leads to the notions of weakly sparse, approximately sparse and sparse graphs, as well as graphs which satisfy a strong isoperimetric inequality.

entreat

chemoprevention

Harmonic Functions and Caccioppoli TheoryThe key tool for all of these results are variants of the Caccioppoli inequality which are established in Section 12.1. Roughly speaking, such inequalities allow us to estimate the energy of . times a cutoff function by . times the energy of the cutoff function.

衍生

Spectral BoundsIn this section we prove an analogue to Cheeger’s famous theorem on Riemannian manifolds. This result relates an isoperimetric constant, called the Cheeger constant, to the bottom of the spectrum.