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Symmetry and complexity are the focus of a selection of outstanding papers, ranging from pure Mathematics and Physics to Computer Science and Engineering applications. This collection is based around fundamental problems arising from different fields, but all of them have the same task, i.e. breaking the complexity by the symmetry. In particular, in this Issue, there is an interesting paper dealing with circular multilevel systems in the frequency domain, where the analysis in the frequency domain gives a simple view of the system. Searching for symmetry in fractional oscillators or the analysis of symmetrical nanotubes are also some important contributions to this Special Issue. More papers, dealing with intelligent prognostics of degradation trajectories for rotating machinery in engineering applications or the analysis of Laplacian spectra for categorical product networks, show how this subject is interdisciplinary, i.e. ranging from theory to applications. In particular, the papers by Lee, based on the dynamics of trapped solitary waves for special differential equations, demonstrate how theory can help us to handle a practical problem. In this collection of papers, although encompassing various different fields, particular attention has been paid to the common task wherein the complexity is being broken by the search for symmetry.

History of engineering & technology --- fractional differential equations --- fractional oscillations (vibrations) --- fractional dynamical systems --- nonlinear dynamical systems --- harmonic wavelet --- filtering --- multilevel system --- forced Korteweg-de Vries equation --- trapped solitary wave solutions --- numerical stability --- two bumps or holes --- finite difference method --- Laplacian spectra --- categorical product --- Kirchhoff index --- global mean-first passage time --- spanning tree --- degradation trajectories prognostic --- asymmetric penalty sparse decomposition (APSD) --- rolling bearings --- wavelet neural network (WNN) --- recursive least squares (RLS) --- health indicators --- first multiple Zagreb index --- second multiple Zagreb index, hyper-Zagreb index --- Zagreb polynomials --- Nanotubes

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Graph theory is an important area of applied mathematics with a broad spectrum of applications in many fields. This book results from aSpecialIssue in the journal Mathematics entitled “Graph-Theoretic Problems and Their New Applications”. It contains 20 articles covering a broad spectrum of graph-theoretic works that were selected from 151 submitted papers after a thorough refereeing process. Among others, it includes a deep survey on mixed graphs and their use for solutions ti scheduling problems. Other subjects include topological indices, domination numbers of graphs, domination games, contraction mappings, and neutrosophic graphs. Several applications of graph theory are discussed, e.g., the use of graph theory in the context of molecular processes.

Zagreb indices --- n/a --- generating function --- mitotic cell cycle --- Mycielskian graph --- evolution theory --- grids --- “partitions” of wheel graph --- generalized hypertree --- connectivity --- single-valued neutrosophic graph --- degree of a vertex --- domination game --- interval-valued intuitionistic fuzzy graph --- directed cycle --- makespan criterion --- total-colored graph --- bipartite matching extendable graph --- stochastic convergence --- bipartite neutrosophic graph --- signless Laplacian --- complete neutrosophic graph --- k-trees --- enhanced hypercube --- b-metric space --- resistance distance --- Wiener index --- mixed graph --- line graph --- NP-hard --- generalized first Zagreb index --- inverse degree index --- sum lordeg index --- Edge Wiener --- chromatic polynomial --- degree of vertex --- complement neutrosophic graph --- graphic contraction mappings --- embedding --- Cartesian product --- k-rainbow domination number --- distance between two vertices --- evolution algebra --- k-rainbow dominating function --- PI index --- subtree --- component --- competition-independence game --- interval-valued fuzzy graph --- b-metric-like space --- induced matching extendable --- edge coloring --- degree of edge --- approximation methods --- chromatic index --- join of graphs --- genetic algorithm --- hypergraph --- edge congestion --- complement --- polynomials in graphs --- vertex coloring --- interval-valued neutrosophic graph --- spanning tree --- Kempe chain --- general contractive mappings --- DD index --- wireless multihop network and social network --- distance --- evolutionary approach --- complexity analysis --- neutrosophic graph --- Kempe-locking --- wheel graph --- Birkhoff diamond --- domination number --- k-extendable --- degree-Kirchhoff index --- adjacent matrix --- perfect matching --- spectral radius --- normalized Laplacian --- corona product --- road transport network --- extremal values --- bound --- chromatic number --- graph coloring --- combinatorial optimization --- reformulated Zagreb indices --- wirelength --- intuitionistic fuzzy graph --- unit-time scheduling --- fan graph --- "partitions" of wheel graph