Graph adjacency and Laplacian generator conventions

Let $V$ be a finite vertex set with a fixed order and let $w:V\times V\to [0,\mathrm{\infty })$ be symmetric. Self-loops are allowed and their weight is counted once in each row sum. On ${\mathbb{C}}^V$ with ordered basis $(e_v{)}_{v\in V}$, define the weighted adjacency matrix by $A_{uv}=w(u,v)$, the degree $d(v)={\sum }_{u\in V}w(v,u)$, and $D_{vv}=d(v)$ with all off-diagonal entries zero. The positive Laplacian is $L_{+}=D-A$; the other common sign convention is $L_{-}=A-D=-L_{+}$. A walk generator records the chosen edge weights, loop convention, basis order, and one of these signs. It is not a discrete transition matrix.

When every $d(v)>0$, the normalized adjacency is the separately defined matrix $A_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}=D^{-1\mathrm{/}2}A{D}^{-1\mathrm{/}2}$. No normalized-adjacency operation is defined here at zero-degree vertices. The matrices $A,D,L_{+},L_{-}$ are Hermitian: $A_{vu}=w(v,u)=w(u,v)=\overline{A_{uv}}$ and $D$ is real diagonal. Under the positive-degree condition, $D^{-1\mathrm{/}2}$ is also real diagonal, so $A_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}^{\ast }=A_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$. A zero-vertex graph gives the zero coordinate space and empty matrices; a quantum-walk input separately requires a normalized state, so it has no instance on that space.