The blowup-polynomial of a metric space: connections to stable polynomials, graphs and their distance spectra
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| Format: | Preprint |
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2021
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| _version_ | 1866918446169587712 |
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| author | Choudhury, Projesh Nath Khare, Apoorva |
| author_facet | Choudhury, Projesh Nath Khare, Apoorva |
| contents | 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 hitherto unstudied delta-matroid to each metric space $X$; we produce another novel delta-matroid for each tree, which interestingly does not generalize to all graphs.
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 three further results: (a) 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. (b) 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 $G$. (c) We obtain a novel characterization of complete multipartite graphs, as precisely those for which the "homogenization at $-1$" of $p_G({\bf n})$ is real-stable (equivalently, Lorentzian, or strongly/completely log-concave), if and only if the normalization of $p_G(-{\bf n})$ is strongly Rayleigh. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_12111 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The blowup-polynomial of a metric space: connections to stable polynomials, graphs and their distance spectra Choudhury, Projesh Nath Khare, Apoorva Metric Geometry Classical Analysis and ODEs Combinatorics 26C10, 05C31 (primary), 30L05, 15A15, 05C12, 05C50 (secondary) 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 hitherto unstudied delta-matroid to each metric space $X$; we produce another novel delta-matroid for each tree, which interestingly does not generalize to all graphs. 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 three further results: (a) 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. (b) 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 $G$. (c) We obtain a novel characterization of complete multipartite graphs, as precisely those for which the "homogenization at $-1$" of $p_G({\bf n})$ is real-stable (equivalently, Lorentzian, or strongly/completely log-concave), if and only if the normalization of $p_G(-{\bf n})$ is strongly Rayleigh. |
| title | The blowup-polynomial of a metric space: connections to stable polynomials, graphs and their distance spectra |
| topic | Metric Geometry Classical Analysis and ODEs Combinatorics 26C10, 05C31 (primary), 30L05, 15A15, 05C12, 05C50 (secondary) |
| url | https://arxiv.org/abs/2105.12111 |