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Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group.
strongly regular graph  automorphism group  orbit matrix  binary polyhedral group  icosahedron  dodecahedron  600cell  Electric multiple unit trains  highlevel maintenance planning  time window  0–1 programming model  particle swarm algorithm  fixed point  splitquaternion  quadratic polynomial  splitoctonion  neutrosophic set  neutrosophic rough set  pessimistic (optimistic) multigranulation neutrosophic approximation operators  complete lattice  rough set  matroid  operator  attribute reduction  graded rough sets  rough intuitionistic fuzzy sets  dominance relation  logical conjunction operation  logical disjunction operation  multigranulation  planar point set  convex polygon  disjoint holes  fuzzy logic  pseudoBCI algebra  quasimaximal element  KGunion  quasialternating BCKalgebra  quality function deployment  engineering characteristics  group decision making  2tuple  metro station  emergency routes  graph partitioning  graph clustering  invariant measures  partition comparison  finite automorphism groups  graph automorphisms  Fuzzy sets  ring  normed space  fuzzy normed ring  fuzzy normed ideal  fuzzy implication  quantum Balgebra  qfilter  quotient algebra  basic implication algebra  Detour–Harary index  maximum  unicyclic  bicyclic  cacti  threeway decisions  intuitionistic fuzzy sets  multigranulation rough intuitionistic fuzzy sets  granularity importance degree  complexity  Chebyshev polynomials  gear graph  pyramid graphs  edge detection  Laplacian operation  regularization  parameter selection  performance evaluation  aggregation operator  triangular norm  ?convex set  atombond connectivity index  geometric arithmetic index  line graph  generalized bridge molecular graph  graceful labeling  edge graceful labeling  edge even graceful labeling  polar grid graph  graph  good drawing  crossing number  join product  cyclic permutation  nonlinear  synchronized  linear discrete  chaotic system  algorithm  generalized permanental polynomial  coefficient  copermanental  isoperimetric number  random graph  intersection graph  social network  Abel–Grassmann’s groupoid (AGgroupoid)  Abel–Grassmann’s group (AGgroup)  involution AGgroup  commutative group  filter  graceful labeling  edge even graceful labeling  cylinder grid graph  selective maintenance  multistate system  human reliability  optimization  genetic algorithm  hypernearring  multitransformation  embedding  distance matrix (spectrum)  distance signlees Laplacian matrix (spectrum)  (generalized) distance matrix  spectral radius  transmission regular graph  graph  good drawing  crossing number  join product  cyclic permutation  cyclic associative groupoid (CAgroupoid)  cancellative  variant CAgroupoids  decomposition theorem  construction methods
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The present book contains 20 articles collected from amongst the 53 total submitted manuscripts for the Special Issue “Fuzzy Sets, Fuzzy Loigic and Their Applications” of the MDPI journal Mathematics. The articles, which appear in the book in the series in which they were accepted, published in Volumes 7 (2019) and 8 (2020) of the journal, cover a wide range of topics connected to the theory and applications of fuzzy systems and their extensions and generalizations. This range includes, among others, management of the uncertainty in a fuzzy environment; fuzzy assessment methods of humanmachine performance; fuzzy graphs; fuzzy topological and convergence spaces; bipolar fuzzy relations; type2 fuzzy; and intuitionistic, intervalvalued, complex, picture, and Pythagorean fuzzy sets, soft sets and algebras, etc. The applications presented are oriented to finance, fuzzy analytic hierarchy, green supply chain industries, smart health practice, and hotel selection. This wide range of topics makes the book interesting for all those working in the wider area of Fuzzy sets and systems and of fuzzy logic and for those who have the proper mathematical background who wish to become familiar with recent advances in fuzzy mathematics, which has entered to almost all sectors of human life and activity.
fuzzy analytic hierarchy process  decisionmaking  partial consensus  posterior aggregation  fuzzy topology  fuzzy convergence  latticevalued convergence  ?convergence space  relative topologicalness  ptopologcalness  diagonal condition  neighborhood condition  fuzzy sets (FSs)  uncertainty  center of gravity (COG) defuzzification technique  triangular fuzzy numbers (TFNs)  grey numbers (GNs)  fuzzy relation equations (FRE)  grade point average (GPA) index  residuated lattice  soft set  filter  tfilter  tintsoft filter  BCK/BCIalgebra  (?,?)US set  (?,?)US subalgebra  (?,?)US (closed) ideal  (?,?)US commutative ideal  fuzzy topology  fuzzy convergence  ?convergence space  regularity  intuitionistic fuzzy set  similarity  counterintuitive  fuzzy graph  picture fuzzy graph  social network  direct product  lexicographic product  strong product  Intervalvalued complex fuzzy sets  distance measures  rotational invariance  reflectional invariance  dynamic multiple attribute decision making  fuzzy regression  interval type2 fuzzy sets  fuzzy topology  fuzzy convergence  ?convergence  diagonal condition  Lcotopological space  Ltopological space  degree of Lcontinuity  degree of Lclosedness  degree of Lopenness  performance management  green supply chain  decision making  nonlinear mathematical modeling  FMCG industries  fuzzy DEMATEL  bipolar fuzzy relation  bipolar fuzzy reflexive (resp., symmetric and transitive) relation  bipolar fuzzy equivalence relation  bipolar fuzzy partition  (a, b)level set  fuzzy logic  finance  banking  banking crisis management  fuzzy number  ?limit set  periodic point  Zadeh’s extension  smart health  fuzzy collaborative intelligence  fuzzy analytic hierarchy process  suitability  picture fuzzy numbers  interaction operations  partitioned Heronian  aggregation operators  hotel selection  Pythagorean fuzzy set  Dempster–Shafer evidence theory  basic probability assignment  medical diagnosis  intervalvalued fuzzy soft set  intervalvalued fuzzy soft topology  intervalvalued fuzzy soft point  intervalvalued fuzzy soft neighborhood  intervalvalued fuzzy soft quasineighborhood  intervalvalued fuzzy soft separation axioms
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Neutrosophy (1995) is a new branch of philosophy that studies triads of the form (,
generalized aggregation operators  interval neutrosophic set (INS)  multiattribute decision making (MADM)  Choquet integral  fuzzy measure  clustering algorithm  neutrosophic association rule  data mining  neutrosophic sets  big data  analytic hierarchy process (AHP)  SWOT analysis  multicriteria decisionmaking (MCDM) techniques  neutrosophic set theory  neutrosophic clustering  image segmentation  neutrosophic cmeans clustering  region growing  dermoscopy  skin cancer  neutosophic extended triplet subgroups  neutrosophic triplet cosets  neutrosophic triplet normal subgroups  neutrosophic triplet quotient groups  shopping mall  photovoltaic plan  decisionmaking trial and evaluation laboratory (DEMATEL)  intervalvalued neutrosophic set  extended ELECTRE III  symmetry  single valued neutrosophic set (SVNS)  neutrosophic multiset (NM)  single valued neutrosophic multiset (SVNM)  cosine measure  multiple attribute decisionmaking  LNGPBM operator  LNGWPBM operator  Linguistic neutrosophic sets  generalized partitioned Bonferroni mean operator  multiple attribute group decisionmaking (MAGDM)  pseudoBCI algebra  hesitant fuzzy set  neutrosophic set  filter  action learning  school administrator  SVM  neutrosophic classification  neutrosophic set  soft set  totally dependentneutrosophic set  totally dependentneutrosophic soft set  generalized De Morgan algebra  complex neutrosophic set  complex neutrosophic graph  fuzzy graph  matrix representation  neutrosophic triplet groups  semigroup  semineutrosophic triplets  classical group of neutrosophic triplets  Ssemigroup of neutrosophic triplets  pseudo primitive elements  neutrosophic sets (NSs)  interval neutrosophic numbers (INNs)  exponential operational laws of interval neutrosophic numbers  interval neutrosophic weighted exponential aggregation (INWEA) operator  multiple attribute decision making (MADM)  typhoon disaster evaluation  simplified neutrosophic linguistic numbers  cloud model  Maclaurin symmetric mean  multicriteria decisionmaking  neutrosophy  DSmT  decisionmaking algorithms  robotic dexterous hands  grasping configurations  grasp type  generalized group  neutrosophic triplet set  neutrosophic triplet group  group  neutrosophic cubic set  neutrosophic cubic graphs  applications of neutrosophic cubic graphs  singlevalued neutrosophic multisets  medical diagnosis  probabilistic rough sets over two universes  threeway decisions  similarity measures  neutrosophic cubic set  decisionmaking  soft sets  support soft sets  interval valued neutrosophic support soft sets  sustainable supplier selection problems (SSSPs)  analytic network process  interdependency of criteria  TOPSIS  neutrosophic set  2inglevalued neutrosophic set  Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS)  integrated weight  maximizing deviation  multiattribute decisionmaking (MADM)  neutrosophic triplet set (NTS)  partial metric spaces (PMS)  fixed point theory (FPT)  neutrosophic triplet  quasi neutrosophic triplet loop  quasi neutrosophic triplet group  BEalgebra  CIalgebra  fuzzy time series  forecasting  twofactor fuzzy logical relationship  multivalued neutrosophic set  Hamming distance  neutrosophic set  prioritized operator  Muirhead mean  multicriteria decisionmaking  aggregation operators  dual aggregation operators  neutrosophic triplet group (NTG)  NTsubgroup  homomorphism theorem  weak commutative neutrosophic triplet group  neutrosophic rough set  MGNRS  dual domains  inclusion relation  decisionmaking  neutromonomorphism  neutroepimorphism  neutroautomorphism  fundamental neutrohomomorphism theorem  first neutroisomorphism theorem  and second neutroisomorphism theorem  linear and nonlinear neutrosophic number  deneutrosophication methods  neutrosophic set  bipolar fuzzy set  neutrosophic bipolar fuzzy set  neutrosophic bipolar fuzzy weighted averaging operator  similarity measure  algorithm  multiple attribute decision making problem  neutrosophic duplets  semigroup  neutrosophic triplet groups  neutrosophic set  fault diagnosis  normal distribution  defuzzification  simplified neutrosophic weighted averaging operator  (commutative) ideal  generalized neutrosophic set  generalized neutrosophic ideal  commutative generalized neutrosophic ideal  linguistic neutrosophic sets  multicriteria group decisionmaking  power aggregation operator  extended TOPSIS method  probabilistic singlevalued (interval) neutrosophic hesitant fuzzy set  multiattribute decision making  aggregation operator  quasigroup  loop  BCIalgebra  BolMoufang  quasi neutrosophic loops  Fenyves identities  Gmetric  neutrosophic Gmetric  neutrosophic sets  clustering  neutrosophic big data  neutrosophic logic  aggregation operator  complement  intersection  membership  neutrosophic soft set  NC power dual MM operator (NCPDMM) operator  NCPMM operator  MADM  MM operator  Neutrosophic cubic sets  PA operator  interval neutrosophic sets  Bonferroni mean  power operator  multiattribute decision making (MADM)  multiple attribute group decision making (MAGDM)  2tuple linguistic neutrosophic sets (2TLNSs)  TODIM model  2TLNNs TODIM method  construction project  MCGDM problems  triangular fuzzy neutrosophic sets (TFNSs)  VIKOR model  TFNNs VIKOR method  potential evaluation  emerging technology commercialization  Qlinguistic neutrosophic variable set  vector similarity measure  cosine measure  Dice measure  Jaccard measure  decision making  inclusion relation  neutrosophic rough set  multiattribute group decisionmaking (MAGDM)  multigranulation neutrosophic rough set (MNRS)  two universes  single valued trapezoidal neutrosophic number  multicriteria group decision making  possibility degree  power aggregation operators  LAsemihypergroups  neutrosophic triplet set  neutrohomomorphism  algorithm  decision making  expert set  generalized neutrosophic set  neutrosophic sets  Qneutrosophic  soft sets  simplified neutrosophic sets (SNSs)  interval number  dependent degree  multicriteria group decisionmaking (MCGDM)  computability  oracle Turing machines  neutrosophic sets  neutrosophic logic  recursive enumerability  oracle computation  criterion functions  neutrosophic computation  neutrosophic logic  quantum computation  computation  logic
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Neutrosophy (1995) is a new branch of philosophy that studies triads of the form (,
generalized aggregation operators  interval neutrosophic set (INS)  multiattribute decision making (MADM)  Choquet integral  fuzzy measure  clustering algorithm  neutrosophic association rule  data mining  neutrosophic sets  big data  analytic hierarchy process (AHP)  SWOT analysis  multicriteria decisionmaking (MCDM) techniques  neutrosophic set theory  neutrosophic clustering  image segmentation  neutrosophic cmeans clustering  region growing  dermoscopy  skin cancer  neutosophic extended triplet subgroups  neutrosophic triplet cosets  neutrosophic triplet normal subgroups  neutrosophic triplet quotient groups  shopping mall  photovoltaic plan  decisionmaking trial and evaluation laboratory (DEMATEL)  intervalvalued neutrosophic set  extended ELECTRE III  symmetry  single valued neutrosophic set (SVNS)  neutrosophic multiset (NM)  single valued neutrosophic multiset (SVNM)  cosine measure  multiple attribute decisionmaking  LNGPBM operator  LNGWPBM operator  Linguistic neutrosophic sets  generalized partitioned Bonferroni mean operator  multiple attribute group decisionmaking (MAGDM)  pseudoBCI algebra  hesitant fuzzy set  neutrosophic set  filter  action learning  school administrator  SVM  neutrosophic classification  neutrosophic set  soft set  totally dependentneutrosophic set  totally dependentneutrosophic soft set  generalized De Morgan algebra  complex neutrosophic set  complex neutrosophic graph  fuzzy graph  matrix representation  neutrosophic triplet groups  semigroup  semineutrosophic triplets  classical group of neutrosophic triplets  Ssemigroup of neutrosophic triplets  pseudo primitive elements  neutrosophic sets (NSs)  interval neutrosophic numbers (INNs)  exponential operational laws of interval neutrosophic numbers  interval neutrosophic weighted exponential aggregation (INWEA) operator  multiple attribute decision making (MADM)  typhoon disaster evaluation  simplified neutrosophic linguistic numbers  cloud model  Maclaurin symmetric mean  multicriteria decisionmaking  neutrosophy  DSmT  decisionmaking algorithms  robotic dexterous hands  grasping configurations  grasp type  generalized group  neutrosophic triplet set  neutrosophic triplet group  group  neutrosophic cubic set  neutrosophic cubic graphs  applications of neutrosophic cubic graphs  singlevalued neutrosophic multisets  medical diagnosis  probabilistic rough sets over two universes  threeway decisions  similarity measures  neutrosophic cubic set  decisionmaking  soft sets  support soft sets  interval valued neutrosophic support soft sets  sustainable supplier selection problems (SSSPs)  analytic network process  interdependency of criteria  TOPSIS  neutrosophic set  2inglevalued neutrosophic set  Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS)  integrated weight  maximizing deviation  multiattribute decisionmaking (MADM)  neutrosophic triplet set (NTS)  partial metric spaces (PMS)  fixed point theory (FPT)  neutrosophic triplet  quasi neutrosophic triplet loop  quasi neutrosophic triplet group  BEalgebra  CIalgebra  fuzzy time series  forecasting  twofactor fuzzy logical relationship  multivalued neutrosophic set  Hamming distance  neutrosophic set  prioritized operator  Muirhead mean  multicriteria decisionmaking  aggregation operators  dual aggregation operators  neutrosophic triplet group (NTG)  NTsubgroup  homomorphism theorem  weak commutative neutrosophic triplet group  neutrosophic rough set  MGNRS  dual domains  inclusion relation  decisionmaking  neutromonomorphism  neutroepimorphism  neutroautomorphism  fundamental neutrohomomorphism theorem  first neutroisomorphism theorem  and second neutroisomorphism theorem  linear and nonlinear neutrosophic number  deneutrosophication methods  neutrosophic set  bipolar fuzzy set  neutrosophic bipolar fuzzy set  neutrosophic bipolar fuzzy weighted averaging operator  similarity measure  algorithm  multiple attribute decision making problem  neutrosophic duplets  semigroup  neutrosophic triplet groups  neutrosophic set  fault diagnosis  normal distribution  defuzzification  simplified neutrosophic weighted averaging operator  (commutative) ideal  generalized neutrosophic set  generalized neutrosophic ideal  commutative generalized neutrosophic ideal  linguistic neutrosophic sets  multicriteria group decisionmaking  power aggregation operator  extended TOPSIS method  probabilistic singlevalued (interval) neutrosophic hesitant fuzzy set  multiattribute decision making  aggregation operator  quasigroup  loop  BCIalgebra  BolMoufang  quasi neutrosophic loops  Fenyves identities  Gmetric  neutrosophic Gmetric  neutrosophic sets  clustering  neutrosophic big data  neutrosophic logic  aggregation operator  complement  intersection  membership  neutrosophic soft set  NC power dual MM operator (NCPDMM) operator  NCPMM operator  MADM  MM operator  Neutrosophic cubic sets  PA operator  interval neutrosophic sets  Bonferroni mean  power operator  multiattribute decision making (MADM)  multiple attribute group decision making (MAGDM)  2tuple linguistic neutrosophic sets (2TLNSs)  TODIM model  2TLNNs TODIM method  construction project  MCGDM problems  triangular fuzzy neutrosophic sets (TFNSs)  VIKOR model  TFNNs VIKOR method  potential evaluation  emerging technology commercialization  Qlinguistic neutrosophic variable set  vector similarity measure  cosine measure  Dice measure  Jaccard measure  decision making  inclusion relation  neutrosophic rough set  multiattribute group decisionmaking (MAGDM)  multigranulation neutrosophic rough set (MNRS)  two universes  single valued trapezoidal neutrosophic number  multicriteria group decision making  possibility degree  power aggregation operators  LAsemihypergroups  neutrosophic triplet set  neutrohomomorphism  algorithm  decision making  expert set  generalized neutrosophic set  neutrosophic sets  Qneutrosophic  soft sets  simplified neutrosophic sets (SNSs)  interval number  dependent degree  multicriteria group decisionmaking (MCGDM)  computability  oracle Turing machines  neutrosophic sets  neutrosophic logic  recursive enumerability  oracle computation  criterion functions  neutrosophic computation  neutrosophic logic  quantum computation  computation  logic  n/a
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