Discrete Dirichlet energy ($\mathcal{E}(f,f)$)

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.