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