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Abstract Motivated by the study of genomes evolving by reversals, we consider pseudograph transformations known as "pressing sequences". In particular, we address the question of when a graph has precisely one pressing sequence resulting in the empty graph, thus answering an question from Cooper and Davis (2015) [13]. We characterize such "uniquely pressable" graphs, count the number of them on a given number of vertices, and provide a polynomial-time recognition algorithm. We conclude with several open questions. [ABSTRACT FROM AUTHOR]

Graph eigenvalues are examples of totally real algebraic integers, i.e. roots of real-rooted monic polynomials with integer coefficients. Conversely, the fact that every totally real algebraic integer occurs as an eigenvalue of some finite graph is a deep and remarkable result, conjectured forty years ago by Hoffman, and proved seventeen years later by Estes. This short paper provides an independent and elementary proof of a stronger statement, namely that the graph may actually be chosen to be a tree. As a by-product, our result implies that the atoms of the limiting spectrum of n × n symmetric matrices with independent Bernoulli ( c n ) entries are exactly the totally real algebraic integers. This settles an open problem raised by Ben Arous (2010). [ABSTRACT FROM AUTHOR]