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This paper focuses on the well-known problem due to Stanley of whether two non-isomorphic trees can have the same U -polynomial (or, equivalently, the same chromatic symmetric function). We consider the U k -polynomial, which is a restricted version of U -polynomial, and construct, for any given k , non-isomorphic trees with the same U k -polynomial. These trees are constructed by...
Given a graph G in which each edge fails independently with probability q ∈ [ 0 , 1 ] , the all-terminal reliability of G is the probability that all vertices of G can communicate with one another, that is, the probability that the operational edges span the graph. The all-terminal reliability is a polynomial in q whose roots (all-terminal reliability roots) were conjectured to have modulus...
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