*Submitted Paper*

**Inserted:** 28 may 2001

**Last Updated:** 18 apr 2003

**Pages:** 20

**Year:** 2001

**Abstract:**

The paper deals with the regularity properties of solutions to elliptic difference equation on a graph $G$ under suitable hypotheses on the geometry of the latter, namely a doubling condition and a PoincarÃ¨ inequality involving discrete derivatives. The difference equation we deal with is of the kind $\sum_{x\sim y}a(x,y)(u(y)-u(x)) + c(x)u(x)=f(x)$ where $x\sim y$ means that the vertices $x$ and $y$ of $G$ are connected each other in the graph structure. The main hypotheses on the coefficients of the equation is that $a$ is symmetric in its arguments and positive. A Harnack inequality for positive solutions is proved, thus giving an HÃ¶lder estimate on the oscillation of a solution in term of suitable $L^p$ norms of the coefficients.

**Keywords:**
graph, elliptic, Harnack, regularity

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