The blowup-polynomial of a metric space: connections to stable polynomials, graphs and their distance spectra

To every finite metric space $X$, including all connected unweighted graphs with the minimum edge-distance metric, we attach an invariant that we call its blowup-polynomial $p_X({ n_x : x in X })$. This is obtained from the blowup $X[{bf n}]$ - which contains $n_x$ copies of each point $x$ - by computing the determinant of the distance matrix of $X[{bf n}]$ and removing an exponential factor. We prove that as a function of the sizes $n_x$, $p_X({bf n})$ is a polynomial, is multi-affine, and is real-stable. This naturally associates a delta-matroid to each metric space $X$ (and another delta-matroid to every tree), which also seem to be hitherto unexplored. We moreover show that the homogenization at $-1$ of $p_X({bf n})$ is Lorentzian (or strongly/completely log-concave), if and only if the normalization of $p_X(-{bf n})$ is strongly Rayleigh, if and only if a modification of the distance matrix of $X$ is positive semidefinite. We next specialize to the case of $X = G$ a connected unweighted graph - so $p_G$ is partially symmetric in ${ n_v : v in V(G) }$ - and show two further results: (a) We show that the univariate specialization $u_G(x) := p_G(x,dots,x)$ is a transform of the characteristic polynomial of the distance matrix $D_G$; this connects the blowup-polynomial of $G$ to the well-studied distance spectrum of $G$. (b) We show that the polynomial $p_G$ is indeed a graph invariant, in that $p_G$ and its symmetries recover the graph $G$ and its isometries, respectively.