Abstract:
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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