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This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view.

Zagreb indices --- domination --- Devaney chaos --- rotationally-symmetric convex polytopes --- topologically mixing --- geometric arithmetic index --- connectivity --- Metric dimension --- atom-bond connectivity ABC index --- distinguishing number --- gear graph --- titanium difluoride --- algorithm --- generalized gear graph --- general Randi? index --- cuprite --- atom bond connectivity index --- binary locating-domination number --- basis --- hypercyclicity --- functigraph --- complete graph --- alpha-boron nanotube --- general randi? index --- geodesics --- inverse degree index --- Hex-Derived Cage networks --- metric dimension --- dominating set --- bipartite graphs --- products of graphs --- geometric-arithmetic GA index --- direct product of graphs --- gromov hyperbolicity --- secure resolving set and secure resolving domination --- topological indices --- graph operators --- resolving set --- harmonic index --- polycyclic aromatic hydrocarbons --- ILP models --- disjointness --- topological transitivity --- Gromov hyperbolicity --- metric basis --- harmonic polynomial

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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

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Discrete optimization is an important area of Applied Mathematics with a broad spectrum of applications in many fields. This book results from a Special Issue in the journal Mathematics entitled ‘Advances and Novel Approaches in Discrete Optimization’. It contains 17 articles covering a broad spectrum of subjects which have been selected from 43 submitted papers after a thorough refereeing process. Among other topics, it includes seven articles dealing with scheduling problems, e.g., online scheduling, batching, dual and inverse scheduling problems, or uncertain scheduling problems. Other subjects are graphs and applications, evacuation planning, the max-cut problem, capacitated lot-sizing, and packing algorithms.

Research & information: general --- Mathematics & science --- forgotten index --- balaban index --- reclassified the zagreb indices --- ABC4 index --- GA5 index --- HDN3(m) --- THDN3(m) --- RHDN3(m) --- degree of vertex --- extended adjacency index --- scheduling with rejection --- machine non-availability --- operator non-availability --- dynamic programming --- FPTAS --- Transportation --- batching scheduling --- total weighted completion time --- unary NP-hard --- approximation algorithm --- bi-criteria scheduling --- online algorithm --- makespan --- maximum machine cost --- competitive ratio --- network optimization --- dynamic flow --- evacuation planning --- contraflow configuration --- partial lane reversals, algorithms and complexity --- logistic supports --- scheduling algorithm --- release-time --- due-date --- divisible numbers --- lateness --- bin packing --- time complexity --- batch scheduling --- linear deterioration --- job families --- Max-cut problem --- combinatorial optimization --- deep learning --- pointer network --- supervised learning --- reinforcement learning --- capacitated lot sizing --- mixed integer formulation --- retail --- inventory --- shortages --- graph --- join product --- crossing number --- cyclic permutation --- arithmetic mean --- combinatorial generation --- method --- algorithm --- AND/OR tree --- Euler-Catalan's triangle --- labeled Dyck path --- ranking algorithm --- unranking algorithm --- Harris hawks optimizer --- load frequency control --- sensitivity analysis --- smart grid --- particle swarm optimization --- genetic algorithm --- meta-heuristics --- packing --- irregular 3D objects --- quasi-phi-function s --- nonlinear optimization --- single-machine scheduling --- minimization of maximum penalty --- dual problem --- inverse problem --- branch and bound --- LNS --- numerical conversion --- RISC --- FPGA --- embedded systems --- scheduling --- job-shop --- makespan criterion --- uncertain processing times --- forgotten index --- balaban index --- reclassified the zagreb indices --- ABC4 index --- GA5 index --- HDN3(m) --- THDN3(m) --- RHDN3(m) --- degree of vertex --- extended adjacency index --- scheduling with rejection --- machine non-availability --- operator non-availability --- dynamic programming --- FPTAS --- Transportation --- batching scheduling --- total weighted completion time --- unary NP-hard --- approximation algorithm --- bi-criteria scheduling --- online algorithm --- makespan --- maximum machine cost --- competitive ratio --- network optimization --- dynamic flow --- evacuation planning --- contraflow configuration --- partial lane reversals, algorithms and complexity --- logistic supports --- scheduling algorithm --- release-time --- due-date --- divisible numbers --- lateness --- bin packing --- time complexity --- batch scheduling --- linear deterioration --- job families --- Max-cut problem --- combinatorial optimization --- deep learning --- pointer network --- supervised learning --- reinforcement learning --- capacitated lot sizing --- mixed integer formulation --- retail --- inventory --- shortages --- graph --- join product --- crossing number --- cyclic permutation --- arithmetic mean --- combinatorial generation --- method --- algorithm --- AND/OR tree --- Euler-Catalan's triangle --- labeled Dyck path --- ranking algorithm --- unranking algorithm --- Harris hawks optimizer --- load frequency control --- sensitivity analysis --- smart grid --- particle swarm optimization --- genetic algorithm --- meta-heuristics --- packing --- irregular 3D objects --- quasi-phi-function s --- nonlinear optimization --- single-machine scheduling --- minimization of maximum penalty --- dual problem --- inverse problem --- branch and bound --- LNS --- numerical conversion --- RISC --- FPGA --- embedded systems --- scheduling --- job-shop --- makespan criterion --- uncertain processing times

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Discrete optimization is an important area of Applied Mathematics with a broad spectrum of applications in many fields. This book results from a Special Issue in the journal Mathematics entitled ‘Advances and Novel Approaches in Discrete Optimization’. It contains 17 articles covering a broad spectrum of subjects which have been selected from 43 submitted papers after a thorough refereeing process. Among other topics, it includes seven articles dealing with scheduling problems, e.g., online scheduling, batching, dual and inverse scheduling problems, or uncertain scheduling problems. Other subjects are graphs and applications, evacuation planning, the max-cut problem, capacitated lot-sizing, and packing algorithms.

forgotten index --- balaban index --- reclassified the zagreb indices --- ABC4 index --- GA5 index --- HDN3(m) --- THDN3(m) --- RHDN3(m) --- degree of vertex --- extended adjacency index --- scheduling with rejection --- machine non-availability --- operator non-availability --- dynamic programming --- FPTAS --- Transportation --- batching scheduling --- total weighted completion time --- unary NP-hard --- approximation algorithm --- bi-criteria scheduling --- online algorithm --- makespan --- maximum machine cost --- competitive ratio --- network optimization --- dynamic flow --- evacuation planning --- contraflow configuration --- partial lane reversals, algorithms and complexity --- logistic supports --- scheduling algorithm --- release-time --- due-date --- divisible numbers --- lateness --- bin packing --- time complexity --- batch scheduling --- linear deterioration --- job families --- Max-cut problem --- combinatorial optimization --- deep learning --- pointer network --- supervised learning --- reinforcement learning --- capacitated lot sizing --- mixed integer formulation --- retail --- inventory --- shortages --- graph --- join product --- crossing number --- cyclic permutation --- arithmetic mean --- combinatorial generation --- method --- algorithm --- AND/OR tree --- Euler–Catalan’s triangle --- labeled Dyck path --- ranking algorithm --- unranking algorithm --- Harris hawks optimizer --- load frequency control --- sensitivity analysis --- smart grid --- particle swarm optimization --- genetic algorithm --- meta-heuristics --- packing --- irregular 3D objects --- quasi-phi-function s --- nonlinear optimization --- single-machine scheduling --- minimization of maximum penalty --- dual problem --- inverse problem --- branch and bound --- LNS --- numerical conversion --- RISC --- FPGA --- embedded systems --- scheduling --- job-shop --- makespan criterion --- uncertain processing times --- n/a --- Euler-Catalan's triangle