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Analysis of vertices plays an important role in Graph Theory and Fuzzy Graph Theory and characterization of different types of vertices is a challenging problem. This book deals with different types of vertices such as eccentric vertices, peripheral vertices, central vertices, complete vertices, cut vertices, interior vertices and boundary vertices in graphs and fuzzy graphs using geodesic and detour distances. The material in this text book is useful and interesting for Mathematics students as well as for other students whose areas of interest include Graphs and/or Fuzzy Graphs.