The k-th power of an n-vertex graph X is the iterated cartesian product of X with itself. The k-th symmetric power of X is the quotient graph of certain subgraph of its k-th power by the natural action of the symmetric group. It is natural to ask if the spectrum of the k-th power -or the spectrum of the k-th symmetric power -is a complete graph invariant for small values of k, for example, for k = O (1) or k = O (log n).In this paper, we answer this question in the negative: we prove that if the well-known 2k-dimensional Weisfeiler-Lehman method fails to distinguish two given graphs, then their k-th powers -and their k-th symmetric powers -are cospectral. As it is well known, there are pairs of non-isomorphic n-vertex graphs which are not distinguished by the k-dim WL method, even for k = Ω(n). In particular, this shows that for each k, there are pairs of non-isomorphic n-vertex graphs with cospectral k-th (symmetric) powers.

We study energy transfer in a "resonant duet" -a resonant quartet where symmetries support a reduced subsystem with only two degrees of freedom -where one mode is forced by white noise and the other is damped. We consider a physically motivated family of nonlinear damping forms, and investigate their effect on the dynamics of the system. A variety of statistical steady-states arise in different parameter regimes, including intermittent bursting phases, non-equilibrium states highly constrained by slaving among amplitudes and phases, and Gaussian and non-Gaussian quasi-equilibrium regimes. All of this can be understood analytically using asymptotic techniques for stochastic differential equations.

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