Geodesic
On stability of weighted spanning tree degree enumerators
Izvestiya. Mathematics , Tome 89 (2025) no. 1, pp. 106-124

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In [1] it was shown that the degree (vertex) spanning tree enumerator polynomial of a connected graph $G$ is a real stable polynomial (that is, it does not vanish if all the variables have positive imaginary parts) if and only if $G$ is a distance-hereditary graph. We prove a similar characterization for weighted graphs. With the help of this generalization, define the class of weighted distance-hereditary graphs.
Keywords: weighted graphs, spanning trees, real stable polynomials, distance-hereditary graphs. 
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P. K. Prozorov; D. D. Cherkashin. On stability of weighted spanning tree degree enumerators. Izvestiya. Mathematics , Tome 89 (2025) no. 1, pp. 106-124. http://geodesic.mathdoc.fr/item/IM2_2025_89_1_a4/