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