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Abstract: The subject of the talk lies at the intersection of combinatorics, linear algebra and complex analysis.
Calculating the number of spanning trees in a graph goes back to the celebrated result of Kirchhoff (1847), who connected the number of trees and the Laplacian matrix. However, as was shown by Cayley (1889), there is a bijection between the summands of (x_1 + x_2 + ... + x_n)^(n-2) and the trees in a complete n-vertex graph (and in particular the number of trees is n^(n-2)). Here a tree T corresponds to a monomial in which degree of x_i is equal to the degree of vertex i in T minus 1. In other words the vertex spanning enumerator polynomial of a complete graph has a linear factorization. We discuss all strengthening of these results that I know, in particular the following result by C., Petrov and Prozorov (2023).
The following statements are equivalent.
2) The vertex spanning enumerator polynomial of G factors into linear terms.
3) The vertex spanning enumerator polynomial of G is real stable (i.e., it does not vanish when substituting any variables from the open upper complex half-plane).
We finish with open questions.
Bio: Dr. Cherkashin is a brilliant expert in Graph Theory, Combinatorics and Dynamical Systems. He graduated from Saint Petersburg State University in 2015, got his PhD in 2018 (Saint Petersburg Department of Steklov Mathematical Institute) and habilitated at IMI BAS in 2024. Being graduated, he has worked at Department of Mathematics and Computer Sciences from Saint Petersburg State University before moving to IMI BAS in 2022.
