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In this paper, we prove bounds for the unique, positive zero of O G (z) := 1 −O G (z) , where O G ( z ) is the socalled orbit polynomial [1]. The orbit polynomial is based on the multiplic ity and cardinalities of the vertex orbits of a graph. In [1] , we have shown that the unique, positive zero δ≤1 of O G (z) can serve as a meaningful measure of graph symmetry. In this paper, we study special graph classes with a specified number of orbits and obtain bounds on the value of δ.

Research on the structural complexity of networks has produced many useful results in graph theory and applied disciplines such as engineering and data analysis. This paper is intended as a further contribution to this area of research. Here we focus on measures designed to compare graphs with respect to symmetry. We do this by means of a novel characteristic of a graph G, namely an ``orbit polynomial.'' A typical term of this univariate polynomial is of the form czn, where c is the number of orbits of size n of the automorphism group of G. Subtracting the orbit polynomial frommore »

Extending the framework of statistical physics to the nonequilibrium setting has led to the discovery of previously unidentified phases of matter, often catalyzed by periodic driving. However, preventing the runaway heating that is associated with driving a strongly interacting quantum system remains a challenge in the investigation of these newly discovered phases. In this work, we utilize a trappedion quantum simulator to observe the signatures of a nonequilibrium driven phase without disorder—the prethermal discrete time crystal. Here, the heating problem is circumvented not by disorderinduced manybody localization, but rather by highfrequency driving, which leads to an expansive time window wheremore »