Foundations
This page recalls standard definitions from simplicial (co)homology and discrete Hodge theory. Nothing here is new; the goal is to fix notation used across the site.
Oriented simplices and cochains
Let $X$ be a finite simplicial complex. For each $k$, let $X_k$ be the set of oriented $k$-simplices. The space of real $k$-cochains is $C^k(X)=\mathbb{R}^{X_k}$, with the standard inner product.
Coboundaries
The coboundary operators $\delta_k:C^k(X)\to C^{k+1}(X)$ are defined by signed incidence: for an oriented $(k+1)$-simplex $\sigma=[v_0,\dots,v_{k+1}]$, $$ (\delta_k f)(\sigma)=\sum_{i=0}^{k+1}(-1)^i f([v_0,\dots,\widehat{v_i},\dots,v_{k+1}]). $$ They satisfy $\delta_{k+1}\delta_k=0$.
Laplacians and Hodge decomposition
The (combinatorial) $k$-Laplacian is $$L_k=\delta_{k-1}\delta_{k-1}^{\mathsf T}+\delta_k^{\mathsf T}\delta_k \;=\;L_k^{\downarrow}+L_k^{\uparrow}.$$ It is positive semidefinite, and $$\ker L_k = \ker(\delta_{k-1}^{\mathsf T})\cap \ker(\delta_k).$$
One has an orthogonal decomposition $$C^k(X)=\operatorname{im}\delta_{k-1}\ \oplus\ \operatorname{im}\delta_k^{\mathsf T}\ \oplus\ \mathcal{H}^k(X),$$ where $\mathcal{H}^k(X)=\ker L_k$ is the harmonic space.
Specializing to graphs and triangles
If $X$ is built from a graph $G=(V,E)$ plus a set of triangles $T$, then $C^0=\mathbb R^V$, $C^1=\mathbb R^{E}$, $C^2=\mathbb R^T$. The operator $\delta_0$ is the signed vertex–edge incidence matrix, while $\delta_1$ encodes edge–triangle incidence. The 1-Laplacian becomes $$L_1(T)=\delta_0\delta_0^{\mathsf T}+\delta_1(T)^{\mathsf T}\delta_1(T).$$
Smallest positive eigenvalue
For a PSD matrix $A$, define $\lambda_{\min}^+(A)$ as the smallest eigenvalue restricted to $(\ker A)^\perp$. The Rayleigh quotient characterization is: $$\lambda_{\min}^+(A)=\min_{x\in(\ker A)^\perp,\ \|x\|=1}\langle Ax,x\rangle.$$