Denoting by $A$ the $n\times n$ adjaceny matrix (with loops contributing $1$ on the diagonal) of a finite graph $\Gamma$ with $n$ vertices, a necessary condition for $A$ to be weighted-regular is the fact that the all $1$ vector $I_n=(1,1,\dots,1)$ of dimension $n$ is in the image of $A$. This is the case if and only if $I_n$ is orthogonal to the kernel $\ker(A)$ of $A$.

If this necessary condition holds, the vector $I_n$ can be written uniquely as $$I_n=\sum_{\lambda\in{\mathrm Spec}(A)\setminus{0}}v_\lambda$$ where the sum is over all distinct non-zero eigenvalues of $A$ with $v_\lambda$ denoting the orthogonal projection of $I_n$ onto the eigenspace of eigenvalue $\lambda$. The graph $\Gamma$ is now weighted-regular if and only if the affine subspace $\sum_{\lambda}\frac{1}{\lambda}v_\lambda+\ker(A)$ (corresponding to the set of preimages of $I_n$) intersects the open cone $({\mathbb R}_{>0})^n$ of vectors having only strictly positive coordinates. This last condition can be checked by linear programming by optimizing an arbitrary linear functional (eg. the coordinate sum) on $\ker(A)$ subject to the condition that all coefficients of the solution are strictly greater to than the corresponding coefficients of $I_n-\sum_{\lambda}\frac{1}{\lambda}v_\lambda$.

In particular, a graph $\Gamma$ with invertible adjacency matrix $A$ is weighted-regular if and only if all coordinates of the vector $\sum_\lambda \frac{1}{\lambda}v_\lambda$ are strictly positive and these coordinates give the unique set of weights (up to multiplication by a positive constant) turning $\Gamma$ into a weighted-regular graph.

This is of course nothing else than Kristall Cantwell's remark, fleshed out.

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Denoting by $A$ the $n\times n$ adjaceny matrix (with loops contributing $1$ on the diagonal) of a finite graph $\Gamma$ with $n$ vertices, the all $1$ vector $I_n=(1,1,\dots,1)$ of dimension $n$ can be written uniquely as $$I_n=\sum_{\lambda\in{\mathrm Spec}(A)\setminus{0}}v_\lambda$$ where the sum is over all distinct non-zero eigenvalues of $A$ with $v_\lambda$ denoting the orthogonal projection of $I_n$ onto the eigenspace of eigenvalue $\lambda$. The graph $\Gamma$ is now weighted-regular if and only if the affine subspace $\sum_{\lambda}\frac{1}{\lambda}v_\lambda+\ker(A)$ intersects the open cone $({\mathbb R}_{>0})^n$ of vectors having only strictly positive coordinates. This last condition can be checked by linear programming by optimizing an arbitrary linear functional (eg. the coordinate sum) on $\ker(A)$ subject to the condition that all coefficients of the solution are strictly greater to the corresponding coefficients of $I_n-\sum_{\lambda}\frac{1}{\lambda}v_\lambda$.

In particular, a graph $\Gamma$ with invertible adjacency matrix $A$ is weighted-regular if and only if all coordinates of the vector $\sum_\lambda \frac{1}{\lambda}v_\lambda$ are strictly positive and these coordinates give the unique set of weights (up to multiplication by a positive constant) turning $\Gamma$ into a weighted-regular graph.

This is of course nothing else than Kristall Cantwell's remark, fleshed out.

1

Denoting by $A$ the $n\times n$ adjaceny matrix (with loops contributing $1$ on the diagonal) of a finite graph $\Gamma$ with $n$ vertices, the all $1$ vector $I_n=(1,1,\dots,1)$ of dimension $n$ can be written uniquely as $$I_n=\sum_{\lambda\in{\mathrm Spec}(A)\setminus{0}}v_\lambda$$ where the sum is over all distinct non-zero eigenvalues of $A$ with $v_\lambda$ denoting the orthogonal projection of $I_n$ onto the eigenspace of eigenvalue $\lambda$. The graph $\Gamma$ is now weighted-regular if and only if the affine subspace $\sum_{\lambda}\frac{1}{\lambda}v_\lambda+\ker(A)$ intersects the open cone $({\mathbb R}_{>0})^n$ of vectors having only strictly positive coordinates. This last condition can be checked by linear programming by optimizing an arbitrary linear functional (eg. the coordinate sum) on $\ker(A)$ subject to the condition that all coefficients of the solution are strictly greater to the corresponding coefficients of $I_n-\sum_{\lambda}\frac{1}{\lambda}v_\lambda$.

In particular, a graph $\Gamma$ with invertible adjacency matrix $A$ is weighted-regular if and only if all coordinates of the vector $\sum_\lambda \frac{1}{\lambda}v_\lambda$ are strictly positive and these coordinates give the unique set of weights (up to multiplication by a positive constant) turning $\Gamma$ into a weighted-regular graph.