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For a finite undirected graph with unit edge conductances, define the Discrete Dirichlet energy of a real-valued function $f$ byEach undirected edge is counted once. More generally, multiply each term by its 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 $\pi (x)=\mathrm{deg}(x)/(2m)$, 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.

Let $v$ be the voltage with $v(a)=1$, $v(b)=0$, and harmonic values at every other vertex. For any other admissible $f$, write $f=v+g$, where $g(a)=g(b)=0$. Its discrete Dirichlet energy expands asDiscrete summation by parts turns the cross term intobecause $v$ is harmonic in the interior and $g$ vanishes at the boundary. Thus $v$ minimizes the energy. Under a unit voltage drop, its energy equals the total current from $a$ to $b$, namely the effective conductance $1/R_{\mathrm{eff}}(a,b)$. Therefore the Dirichlet principle gives

Let the square contain the Euclidean ball of radius $R$, set $r_x=\max (1,\Vert x\Vert )$, and define the logarithmic cutoffThen ${\phi }_0=1$ and $\phi =0$ on the boundary. On an edge at radius comparable to $r$, the mean value theorem gives $|{\phi }_x-{\phi }_y|\le C/(r\log R)$. There are $O(r)$ edges in the annulus of radius $r$, hence the discrete Dirichlet energy satisfiesThis logarithmic cutoff is the discrete manifestation of recurrence in two dimensions.