Graph theory - Exam
  • 1. Graph theory is a branch of mathematics that deals with the study of graphs, which are mathematical structures used to model relationships between objects. A graph consists of a set of vertices, or nodes, which are connected by edges, or links. Graph theory has applications in various fields such as computer science, social network analysis, and operational research. It helps in solving problems related to connectivity, routing, optimization, and more. Overall, graph theory provides a powerful framework for analyzing and understanding complex systems and relationships.

    What is a graph in graph theory?
A) A pie chart
B) A line graph
C) A chart or diagram
D) A mathematical structure consisting of vertices and edges
  • 2. What is a vertex in a graph?
A) A point or node in a graph
B) A line connecting two points in a graph
C) A function in graph theory
D) A path between two vertices
  • 3. What is an edge in a graph?
A) A connection between two vertices
B) A vertex with no connections
C) A loop on a vertex
D) A node's color in a graph
  • 4. What is a path in graph theory?
A) An isolated vertex
B) A disconnected graph
C) A cycle in a graph
D) A sequence of edges that connect a sequence of vertices
  • 5. In a simple graph, can an edge connect a vertex to itself?
A) Yes
B) Sometimes
C) Depends on the number of vertices
D) No
  • 6. What is the degree of a vertex in a graph?
A) The size of the graph
B) The number of vertices in the graph
C) The number of edges incident to the vertex
D) The distance from one vertex to another
  • 7. What is a planar graph?
A) A multigraph
B) A graph with cycles
C) A graph that can be drawn on a plane without any edge intersections
D) A disconnected graph
  • 8. What is a weighted graph?
A) A graph with maximum number of edges
B) A graph in which a number (weight) is assigned to each edge
C) An undirected graph
D) A graph with only one vertex
  • 9. What is an isomorphism between two graphs?
A) A loop on a vertex in both graphs
B) A bijection between their vertex sets that preserves edges
C) Two disconnected graphs
D) The same number of vertices in both graphs
  • 10. What was the title of Leonhard Euler's paper that is considered the first in graph theory?
A) On the Nature of Graphs
B) The Seven Bridges of Königsberg
C) Graph Theory and its Applications
D) Solutio Problematis ad Geometriam Situs Pertinentis
  • 11. What type of graph allows edges to connect a vertex to itself?
A) Simple graph
B) Undirected graph
C) Directed graph
D) Multigraph
  • 12. Who introduced the term 'graph' in the context of mathematics?
A) Dénes Kőnig
B) James Joseph Sylvester
C) Leonhard Euler
D) Arthur Cayley
  • 13. Which problem in graph theory involves coloring regions of a map with four colors such that no two adjacent regions share the same color?
A) Four-color problem
B) Graph connectivity problem
C) Knight's tour problem
D) Seven Bridges problem
  • 14. Who first posed the four-color problem?
A) Augustus De Morgan
B) Peter Tait
C) William Rowan Hamilton
D) Francis Guthrie
  • 15. Who donated all royalties from their textbook on graph theory to fund the Pólya Prize?
A) Arthur Cayley
B) Dénes Kőnig
C) Frank Harary
D) Heinrich Heesch
  • 16. Which mathematician's work on trees linked graph theory with theoretical chemistry?
A) Arthur Cayley
B) Dénes Kőnig
C) Frank Harary
D) Leonhard Euler
  • 17. Who published Kirchhoff's circuit laws in 1845?
A) Gustav Kirchhoff
B) Arthur Cayley
C) Dénes Kőnig
D) Leonhard Euler
  • 18. What is the name of the method published by Heinrich Heesch in 1969 for solving the four-color problem?
A) Discharging method
B) Configuration checking
C) Coloring algorithm
D) Graph reduction
  • 19. Who was influential in the field of graph drawing using linear algebraic methods?
A) W. T. Tutte.
B) Floyd.
C) Euler.
D) Dijkstra.
  • 20. Who is credited with the foundational theorem in extremal graph theory?
A) Szemerédi
B) Erdős
C) Rényi
D) Mantel
  • 21. Which data structure lists the neighbors of each vertex separately?
A) Incidence matrix
B) Adjacency list
C) Adjacency matrix
D) Edge list
  • 22. Which decomposition involves covering each edge exactly twice with cycles?
A) Edge coloring
B) Graph factorization
C) Cycle double cover
D) Arboricity
  • 23. What can graph structures represent in evolutionary biology?
A) Evolutionary trees
B) Species extinction events
C) Genetic mutations
D) Habitat destruction
  • 24. What is an Erdős–Rényi model?
A) A technique for partitioning graphs.
B) A method for finding spanning trees.
C) A model for generating random graphs.
D) An algorithm for graph coloring.
  • 25. What is represented by edges in the context of chemical graph theory?
A) Chemical reactions
B) Bonds
C) Molecules
D) Atoms
  • 26. What is the crossing number for a planar graph?
A) Dependent on the weights assigned to edges.
B) One.
C) Equal to the number of vertices.
D) Zero.
  • 27. Who wrote the first textbook on graph theory, published in 1936?
A) Arthur Cayley
B) Dénes Kőnig
C) Frank Harary
D) Leonhard Euler
  • 28. What is a common method in phonology that uses lattice graphs?
A) Optimality theory
B) Head-driven phrase structure grammar
C) Graph databases
D) Semantic networks
  • 29. Which theorem states that every finite group is the group of symmetries of a finite undirected graph?
A) Sylow's theorem
B) Euler's theorem
C) Paley's theorem
D) Frucht's theorem
  • 30. Who generalized the results of Pólya between 1935 and 1937?
A) Heinrich Heesch
B) Nicolaas Govert de Bruijn
C) Arthur Cayley
D) Frank Harary
  • 31. In the context of porous media, what do edges represent?
A) Smaller channels connecting the pores
B) Solid structures
C) Pores themselves
D) Fluid flow paths
  • 32. What is the decomposition of a graph into as few forests as possible called?
A) Edge coloring
B) Arboricity
C) Graph factorization
D) Cycle double cover
  • 33. What is the name of the problem that involves coloring graphs embedded on surfaces with arbitrary genus?
A) Graph factorization problem
B) Generalized four-color problem
C) Knight's tour problem
D) Graph connectivity problem
  • 34. Which type of graph is used in finite-state morphology?
A) Lattice graphs
B) Tree-based structures
C) Directed graphs
D) Finite-state transducers
  • 35. What is the term for a graph where attributes are associated with vertices and edges, often used to model real-world systems?
A) Semantic network
B) Causal structure
C) Network
D) Graph database
  • 36. Which data structure is often preferred for sparse graphs due to smaller memory requirements?
A) Matrix structures
B) Adjacency matrix
C) List structures
D) Incidence matrix
  • 37. What do vertices represent in graphs modeling porous media?
A) Pores
B) Channels
C) Solids
D) Fluids
  • 38. Which problem involves finding a tree connecting a given set of vertices with minimum total edge weight?
A) Steiner tree
B) Hamiltonian path problem
C) Traveling salesman problem
D) Minimum spanning tree
  • 39. In chemistry, what do vertices represent in a molecular graph?
A) Atoms
B) Bonds
C) Chemical reactions
D) Molecules
  • 40. Which problem involves finding a spanning tree with the minimum total edge weight?
A) Hamiltonian path problem
B) Minimum spanning tree
C) Traveling salesman problem
D) Steiner tree
  • 41. Which organization reflects the usefulness of graph theory to linguistics?
A) Finite-state transducers
B) TextGraphs
C) VerbNet
D) WordNet
  • 42. In which field are graphs used to model networks of communication and data organization?
A) Linguistics
B) Physics
C) Computer science
D) Biology
  • 43. In computational linguistics, what type of network is important for modeling word meaning in terms of related words?
A) Syntactic trees
B) Semantic networks
C) Lattice graphs
D) Graph databases
  • 44. What is the principle that gives tree-based structures in linguistics their expressive power?
A) Compositionality
B) Optimality theory
C) Finite-state transducers
D) Feature structures
  • 45. Which branch of algebra focuses on adjacency matrix and its spectrum in spectral graph theory?
A) Number theory
B) Linear algebra
C) Group theory
D) Combinatorics
  • 46. Which matrix is a diagonal matrix representing the degree of a vertex?
A) Adjacency matrix
B) Incidence matrix
C) Laplacian matrix
D) Degree matrix
  • 47. Who asked for a factory plan minimizing crossings between tracks?
A) Paul Erdős.
B) Karl Menger.
C) Hungarian mathematician Pál Turán.
D) László Lovász.
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