The PHENIX detector is designed to perform a broad study of A-A, p-A, and p-p collisions to investigate nuclear matter under extreme conditions. A wide variety of probes, sensitive to all timescales, are used to study systematic variations with species and energy as well as to measure the spin structure of the nucleon. Designing for the needs of the heavy-ion and polarized-proton programs has produced a detector with unparalleled capabilities. PHENIX measures electron and muon pairs, photons, and hadrons with excellent energy and momentum resolution. The detector consists of a large number of subsystems that are discussed in other papers in this volume. The overall design parameters of the detector are presented. The PHENIX detector is designed to perform a broad study of A-A, p-A, and p-p collisions to investigate nuclear matter under extreme conditions. A wide variety of probes, sensitive to all timescales, are used to study systematic variations with species and energy as well as to measure the spin structure of the nucleon. Designing for the needs of the heavy-ion and polarized-proton programs has produced a detector with unparalleled capabilities. PHENIX measures electron and muon pairs, photons, and hadrons with excellent energy and momentum resolution. The detector consists of a large number of subsystems that are discussed in other papers in this volume. The overall design parameters of the detector are presented. Disciplines Engineering Physics | Physics Comments This is a manuscript of an article from Nuclear Instruments and Methods in Physics Research

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Symmetries are ubiquitous in real networks and often characterize network features and functions. Here we present a generalization of network symmetry called latent symmetry, which is an extension of the standard notion of symmetry. They are defined in terms of standard symmetries in a reduced version of the network. One unique aspect of latent symmetries is that each one is associated with a size, which provides a way of discussing symmetries at multiple scales in a network. We are able to demonstrate a number of examples of networks (graphs) which contain latent symmetry, including a number of real networks. In numerical experiments, we show that latent symmetries are found more frequently in graphs built using preferential attachment, a standard model of network growth, when compared to non-network like (Erdős-Rényi) graphs. Finally we prove that if vertices in a network are latently symmetric, then they must have the same eigenvector centrality, similar to vertices which are symmetric in the standard sense. This suggests that the latent symmetries present in real-networks may serve the same structural and functional purpose standard symmetries do in these networks. We conclude from these facts and observations that latent symmetries are present in real networks and provide useful information about the network potentially beyond standard symmetries as they can appear at multiple scales.

Two emerging topics in graph theory are the study of cospectral vertices of a graph, and the study of isospectral reductions of graphs. In this paper, we prove a fundamental relationship between these two areas, which is that two vertices of a graph are cospectral if and only if the isospectral reduction over these vertices has a nontrivial automorphism. It is well known that if two vertices of a graph are symmetric, i.e. if there exists a graph automorphism permuting these two vertices, then they are cospectral. This paper extends this result showing that any two cospectral vertices are symmetric in some reduced version of the graph. We also prove that two vertices are strongly cospectral if and only if they are cospectral and the isospectral reduction over these two vertices has simple eigenvalues. We further describe how these results can be used to construct new families of graphs with cospectral vertices. *

We extend the theory of equitable decompositions introduced in [2], where it was shown that if a graph has a particular type of symmetry, i.e. a uniform or basic automorphism φ, it is possible to use φ to decompose a matrix M appropriately associated with the graph. The result is a number of strictly smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix M. We show here that a large class of automorphisms, which we refer to as separable, can be realized as a sequence of basic automorphisms, allowing us to equitably decompose M over any such automorphism. We also show that not only can a matrix M be decomposed but that the eigenvectors of M can also be equitably decomposed. Additionally, we prove under mild conditions that if a matrix M is equitably decomposed the resulting divisor matrix, which is the divisor matrix of the associated equitable partition, will have the same spectral radius as the original matrix M. Last, we describe how an equitable decomposition effects the Gershgorin region Γ(M) of a matrix M, which can be used to localize the eigenvalues of M. We show that the Gershgorin region of an equitable decomposition of M is contained in the Gershgorin region Γ(M) of the original matrix. We demonstrate on a real-world network that by a sequence of equitable decompositions it is possible to significantly reduce the size of a matrix' Gershgorin region.