= Discrete Dirichlet energy
{title2=$\mathcal E(f,f)$}
For a finite <undirected graph> with unit <electrical conductances>[edge conductances], define the <Discrete Dirichlet energy> of a real-valued <function> $f$ by
$$
\mathcal E_G(f,f)=\sum_{\{x,y\}\in E}(f(x)-f(y))^2.
$$
Each undirected <edge> is counted once. More generally, multiply each term by its <electrical conductance>[edge conductance]. The energy is nonnegative and vanishes exactly when $f$ is constant on every <connected component>. Large differences across strongly conducting <edges> are expensive; this is why <Discrete Dirichlet energy> appears in <harmonic extension> and electrical-network principles.
Normalization depends on the chain. For <simple random walk> on a finite connected unweighted <graph> with $m$ <edges>, the <Stationary distribution> is $\pi(x)=\deg(x)/(2m)$, so its <Dirichlet form of a Markov chain> is $\mathcal E_P(f,f)=\mathcal E_G(f,f)/(2m)$. On the three-vertex path with values $0,1,0$, the <graph> energy is $2$, whereas the simple-random-walk energy is $1/2$. Stating which normalization is used matters when comparing <spectral gaps> or effective resistances.
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