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