Covariance under a common positive rescaling of weights

Let $A$ be a network on $n\ge 1$ vertices, let $d_0>0$, $J$ the all-ones matrix, $I$ the identity, and ${\mathrm{\Delta }}_A=A-(d_0\mathrm{/}n)J$, ${\mathrm{\Delta }}_L=L_A-d_{0}I+(d_0\mathrm{/}n)J$. The norm is the induced vertex-space operator norm, and $B⪯C$ means that $C-B$ has nonnegative quadratic form. Under $(A,d_0)\mapsto (\lambda A,\lambda d_0)$ for $\lambda >0$, the degrees, Laplacian, both centered matrices, energy and homogeneous rotating-frame vector field all multiply by $\lambda$. Consequently the inequalities $\mathrm{\parallel }{\mathrm{\Delta }}_A\mathrm{\parallel }\le \alpha d_0$ and $c^{-}{d}_{0}I⪯{\mathrm{\Delta }}_L⪯c^{+}{d}_{0}I$ hold before rescaling if and only if they hold afterward with the same $\alpha ,c^{-},c^{+}$. The network and calculus conventions are Finite symmetric networks and loop removal and Loop removal in the weighted Kuramoto calculus; matrix conventions are Finite vertex Hilbert space in the article convention and Self-adjoint order, norm bounds and diagonal tests.