Fundamental Concepts of Graph Theory: Subgraphs and Connectivity
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Questions and Answers

In graph theory, what is an isomorphism between two simple graphs G and H?

  • A bijection f:V(G)→V(H) such that uv is an edge in G if and only if f(u)f(v) is an edge in H (correct)
  • A function that maps the vertices of G to the edges of H
  • A function that maps the edges of G to the edges of H
  • A function that maps the vertices of G to the vertices of H

    • What is a complete graph in graph theory?

  • A graph where each vertex has a different degree
  • A graph where every vertex has a self-loop
  • A graph where no two vertices share an edge
  • A graph where every vertex is connected to every other vertex (correct)

    • What is the Petersen Graph?

  • A graph with 5 vertices connected in a cycle
  • A graph with 10 vertices arranged in a pentagon shape
  • A graph composed of only complete bipartite graphs
  • A graph where the vertices are subsets of a 5-element set and edges are pairs of disjoint 2-element subsets (correct)

    • What is an adjacency matrix in graph theory?

    <p>A matrix showing which vertices are adjacent in a graph</p> Signup and view all the answers

    What does the degree of a vertex in a graph represent?

    <p>The number of edges incident on that vertex</p> Signup and view all the answers

    What does an incidence matrix in graph theory typically represent?

    <p>The relationship between vertices and edges in a graph</p> Signup and view all the answers

    What is a subgraph of a graph?

    <p>A graph with the same vertices and edges as the original graph</p> Signup and view all the answers

    In graph theory, what does it mean for two vertices to be 'adjacent'?

    <p>They share a common endpoint</p> Signup and view all the answers

    What do we call the number of edges incident upon a vertex?

    <p>Degree</p> Signup and view all the answers

    In an adjacency matrix of a graph, what does an entry of '0' represent?

    <p>The vertices are not connected</p> Signup and view all the answers

    How is the incidence matrix of a graph defined?

    <p>A matrix showing which vertices are endpoints of which edges</p> Signup and view all the answers

    What does it mean for a graph to be 'isomorphic' to another?

    <p>The graphs have the same degree sequence</p> Signup and view all the answers

    In graph theory, what is an induced subgraph?

    <p>A subgraph formed by a subset of vertices of the original graph</p> Signup and view all the answers

    Which of the following sets of vertices forms an independent set in a graph?

    <p>{C, D}</p> Signup and view all the answers

    What is the defining characteristic of a cut-edge in a graph?

    <p>It does not belong to any cycles</p> Signup and view all the answers

    Which theorem characterizes a cut-edge based on its presence in cycles?

    <p>Theorem 1.2.14</p> Signup and view all the answers

    If a graph contains a cycle where edge e=(x, y), how does this imply the connectivity of the component containing e?

    <p>The component contains a path from x to y</p> Signup and view all the answers

    How is an induced subgraph different from a regular subgraph?

    <p>An induced subgraph is defined by the subset of vertices and their connections</p> Signup and view all the answers

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