Given a finite simple connected graph \(G = (V, E)\), we introduce a novel invariant which we call its blowup-polynomial \(p_G({n_v : v in V})\). To do so, we compute the determinant of the distance matrix of the graph blowup, obtained by taking \(n_v\) copies of the vertex \(v\), and remove an exponential factor. First: we show that as a function of the sizes \(n_v, p_G\) is a polynomial, is multi-affine, and is real-stable. Second: we show that the multivariate polynomial \(p_G\) is intimately related to the characteristic polynomial \(q_G\) of the distance matrix \(D_G\), and that it fully recovers \(G\) whereas \(q_G\) does not.
Third: we obtain a novel characterization of the complete multipartite graphs, as precisely those whose "homogenized" blowup-polynomials are Lorentzian/strongly Rayleigh. These results also lead to some hitherto unstudied delta-matroids. (Joint with Projesh Nath Choudhury.)
