Regular multigraph and normalized adjacency

A finite $d$-regular adjacency-slot multigraph on $V=\{1,\dots ,n\}$, with $n\ge 1$ and integer $d>0$, is a symmetric matrix $A\in {\mathbb{Z}}_{\ge 0}^{n\times n}$ with every row sum $d$. $A_{uv}$ counts slots from $u$ to $v$; diagonal entries count loop slots. Its normalized adjacency is $M=A\mathrm{/}d$. We use $⟨f,g⟩=n^{-1}{\sum }_{v}f(v)\overline{g(v)}$ unless an unnormalized inner product is explicitly specified.

An adjacency list records the $d$ destinations per vertex, including repetitions. An ordinary undirected edge has two incidence slots paired by reversal; an ordinary loop has two slots at the same vertex. Such a representation requires even diagonal entries. Any symmetric integer matrix can be converted to it by doubling every slot. Uniform directed-slot sampling chooses one of the $nd$ slots; for a reverse-paired graph it induces the uniform distribution on its $nd\mathrm{/}2$ ordinary edges. Connectivity uses positive off-diagonal entries. When $n=1$, the mean-zero subspace is $\{0\}$.