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# harmonic function

A real or complex-valued function $f:V\to\mathbb{R}$ or $f:V\to\mathbb{C}$ defined on the vertices $V$ of a graph $G=(V,E)$ is called *harmonic* at $v\in V$ if its value at $v$ is its average value at the neighbours of $v$:

$f(v)=\frac{1}{\operatorname{deg}(v)}\sum_{{\{u,v\}\in E}}f(u).$ |

It is called harmonic *except on $A$*, for some $A\subseteq V$, if it is harmonic at each $v\in V\setminus A$, and harmonic if it is harmonic at each $v\in V$.

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## Mathematics Subject Classification

05C99*no label found*

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