Graph Theory Basics Quiz
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Questions and Answers

What distinguishes a graph in graph theory from a graph of a function?

  • Graph theory graphs are used for discrete representations, where function graphs deal with continuous data.
  • Graph theory graphs represent numerical values, whereas function graphs show relationships between variables.
  • Graph theory graphs consist of vertices and edges, while function graphs depict curves on a coordinate plane. (correct)
  • Graph theory graphs model relationships between objects, but function graphs represent mathematical formulas.

In graph theory, what defines an edge?

  • A directed path connecting a source to a destination.
  • An unordered pair of vertices. (correct)
  • A single point connecting two other points.
  • A path between two points with a sequence of intermediate points.

If a graph has an edge that connects a vertex to itself, what type of edge is it?

  • Isolated edge
  • Pendant edge
  • Loop (correct)
  • Parallel edge

What term describes multiple edges that connect the same pair of vertices?

<p>Parallel edges (A)</p> Signup and view all the answers

Using the Handshaking Lemma, if a graph has 5 vertices, each with a degree of 4, how many edges are in the graph?

<p>10 (B)</p> Signup and view all the answers

What is a vertex with a degree of zero called?

<p>Isolated vertex (C)</p> Signup and view all the answers

What is a graph called if it has no loops and no parallel edges?

<p>Simple graph (B)</p> Signup and view all the answers

What does the theorem of even number of odd vertices imply in any given graph?

<p>If there are any odd degree vertices, their number must be even. (B)</p> Signup and view all the answers

In a graph, if the sum of degrees of vertices with an odd degree is $X$, and the sum of degrees of vertices with an even degree is $Y$, and the total sum of all degrees is $Z$, which of the following is always true?

<p>$X$ is even, $Y$ is even, and $Z$ is even. (A)</p> Signup and view all the answers

A network of telephone lines and poles is represented by a graph. What does it mean if removing a single edge disconnects the graph?

<p>The edge represents a critical line. (C)</p> Signup and view all the answers

Which type of graph has no edges?

<p>Null Graph (B)</p> Signup and view all the answers

In a complete graph with $n$ vertices, how many edges are there?

<p>$\frac{n(n-1)}{2}$ (D)</p> Signup and view all the answers

A graph where every vertex has the same degree $k$ is called a:

<p>Regular graph (B)</p> Signup and view all the answers

If a graph's vertex set can be divided into two subsets such that every edge connects vertices from different subsets, what type of graph is it?

<p>Bipartite Graph (A)</p> Signup and view all the answers

If H is a subgraph of G and H contains all vertices of G, then H is called a:

<p>Spanning Subgraph (D)</p> Signup and view all the answers

If a graph $G$ has edges and an edge $e$ is removed from them, what kind of subgraph is formed?

<p>Edge Deleted Subgraph (D)</p> Signup and view all the answers

What characterizes a trail in a graph?

<p>It consists of a sequence of edges and vertices with no repeated edges. (D)</p> Signup and view all the answers

Which statement is true regarding a connected graph?

<p>All vertices are part of only one connected component. (D)</p> Signup and view all the answers

What does the vertex induced subgraph G<U> consist of?

<p>Only the vertices in subset U and edges between them. (C)</p> Signup and view all the answers

In graph theory, what is a cycle?

<p>A path that returns to the starting vertex without repeating any others. (B)</p> Signup and view all the answers

How is an edge deleted subgraph G - e4 defined?

<p>A subgraph including all vertices but removing one edge. (D)</p> Signup and view all the answers

Which of the following defines a disconnected graph?

<p>A graph where there is at least one pair of vertices that are not connected. (A)</p> Signup and view all the answers

What is the result of the theorem regarding the complement of a simple graph G?

<p>If G is disconnected, then its complement G is connected. (B)</p> Signup and view all the answers

What is represented by C(u) in a graph?

<p>The collection of all vertices connected to vertex u. (B)</p> Signup and view all the answers

What condition must be true for a graph to be considered connected based on the partition of its vertex set?

<p>There exists at least one edge connecting two vertices from different subsets. (C)</p> Signup and view all the answers

According to the proof of Theorem 4, what happens when m equals zero?

<p>The number of components equals the number of vertices. (B)</p> Signup and view all the answers

What does the proof of Theorem 5 indicate about a graph with exactly two vertices of odd degree?

<p>There exists a path connecting these two vertices. (A)</p> Signup and view all the answers

In the context of disconnected graphs, what conclusion can be drawn if there are no edges connecting vertex sets G₁ and G₂?

<p>The entirety of G is a disjoint union. (D)</p> Signup and view all the answers

What is the relationship between the number of vertices n, components k, and edges m in a simple graph as presented in Theorem 4?

<p>n - k ≤ m. (A)</p> Signup and view all the answers

What is a critical aspect of the proof technique used in Theorem 4?

<p>It uses induction based on the number of edges. (B)</p> Signup and view all the answers

What implication does a graph with k components have on the existence of edges in the graph?

<p>At least n - k edges must exist. (B)</p> Signup and view all the answers

What defines the complement of a graph G?

<p>It includes edges that are present in G but not in G. (B)</p> Signup and view all the answers

Which of the following statements about the components of a graph is true?

<p>For any two vertices in separate components, there can be no path connecting them. (B)</p> Signup and view all the answers

What was concluded by Euler regarding the Konigsberg Seven Bridge Problem?

<p>It is impossible to do so based on the even-odd degree rule. (A)</p> Signup and view all the answers

In a bipartite graph, which statement is accurate regarding the lengths of cycles?

<p>All cycles in the graph must have even length. (D)</p> Signup and view all the answers

What condition must hold true for a graph to be self-complementary?

<p>The number of vertices must be of the form 4k or 4k + 1. (A)</p> Signup and view all the answers

What defines a connected graph?

<p>All pairs of vertices must have at least one path connecting them. (C)</p> Signup and view all the answers

In the context of graph theory, what does W(G) represent?

<p>The number of connected components in graph G. (C)</p> Signup and view all the answers

Which method can show that a graph with even-length cycles is bipartite?

<p>Defining distances from an arbitrary vertex. (A)</p> Signup and view all the answers

What characteristic is unique to a self-complementary graph's vertices?

<p>The number of edges is equal to that of its complement. (D)</p> Signup and view all the answers

Flashcards

End Vertices

An unordered pair of vertices associated with an edge in a graph.

Simple Graph

A graph that has no loops (edges connecting a vertex to itself) and no parallel edges (multiple edges connecting the same two vertices).

Degree of a Vertex

The number of edges incident with a vertex.

Isolated Vertex

A vertex with degree 0, meaning it has no edges connected to it.

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Pendant Vertex

A vertex with degree 1, meaning it has only one edge connected to it.

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Loop

An edge that connects a vertex to itself.

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Parallel Edges

Multiple edges that have the same end vertices.

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Handshaking Lemma

In any graph, the sum of the degrees of all vertices is equal to twice the number of edges.

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Complete Graph

A graph where each vertex is connected to every other vertex by an edge.

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Null Graph

A graph with no edges. All vertices are isolated.

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Regular Graph

A graph where all vertices have the same degree.

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Bipartite Graph

A graph where the vertex set can be divided into two groups, and edges only connect vertices from different groups.

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Subgraph

A graph H is a subgraph of G if all of H's vertices and edges are also in G.

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Spanning Subgraph

A subgraph where all the vertices of the original graph are present, but some edges might be removed.

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Vertex Deleted Subgraph

A subgraph created by removing a single vertex and all its connected edges from the original graph.

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Edge Deleted Subgraph

A subgraph created by removing a single edge from the original graph.

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Vertex Induced Subgraph

A subgraph defined by a specific subset of vertices, including all edges connecting those vertices. G<U> represents the subgraph induced by vertex set 'U'.

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Edge Induced Subgraph

A subgraph determined by a specific subset of edges, comprising the vertices connected by those edges and the edges themselves. G<F> represents the subgraph induced by edge set 'F'.

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Walk

A sequence of vertices and edges, where each edge connects two consecutive vertices. It can repeat vertices and edges.

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Trail

A walk where no edge is repeated. Example: v₁ e₁ v₂ e₅ v₃ e₁₀ v₃ e₅ v₂ e₃ v₅

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Path

A walk with no repeated vertices. Example: v₂ e₄ v₄ e₈ v₃ e₇ v₅ e₆ v₁

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Cycle

A closed walk with no repeated vertices (except the starting and ending vertex). Example: v₁ e₁ v₂ e₂ v₄ e₈ v₃ e₇ v₅ e₆ v₁

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Complement of a disconnected graph is connected

If a graph G is not connected, then its complement G is connected.

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Connected graph - partition

A graph G is connected if and only if for any division of its vertices into two groups, there's an edge connecting a vertex in one group to a vertex in the other.

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Vertices, components, and edges inequality

The number of vertices minus the number of components in a simple graph is less than or equal to the number of edges.

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Path between odd-degree vertices

If a simple graph has exactly two vertices with an odd number of edges connected to them, there must be a path connecting these two vertices.

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Induced subgraph

In a graph, a subgraph induced by a subset of vertices contains all edges present in the original graph that connect vertices within that subset.

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Definition of a connected graph

For a graph to be connected, every vertex must be reachable from every other vertex through a sequence of edges.

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Disjoint union of graphs

A disjoint union of graphs is a combination where no edges connect vertices from different graphs.

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Connected Graph

A graph where for any two vertices, there's at least one path connecting them.

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Disconnected Graph

A graph with at least two separate parts, where vertices in one part have no edges to vertices in another.

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W(G)

The number of connected components in a graph.

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Self-Complementary Graph

A graph that is its own complement. The complement of a graph is obtained by connecting all non-adjacent vertices in the original graph.

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Konigsberg Seven Bridge Problem

A classic problem in graph theory. It involves finding a path through a network of bridges without crossing any bridge more than once.

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Cyclic Graph

A graph that contains a cycle, meaning a closed path that returns to its starting vertex.

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Acyclic Graph

A graph that has no cycles. Each path in the graph has a unique start and end point.

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Study Notes

Graph Theory Introduction

  • Graph theory involves studying vertices and edges
  • Vertices represent objects and edges represent connections between those objects
  • Applications include: acquaintance, chemical bonding, electrical networks, transportation networks, and binary vectors
  • Techniques used include induction, parity, extremality, counting, and inclusion-exclusion methods

Graph Definition

  • A graph G = (V, E) consists of a set of vertices (V) and a set of edges (E)
  • Vertices are the points, and edges connect these points
  • Each edge is associated with an unordered pair of vertices (end vertices)

Elementary Terminologies

  • End vertices of an edge: The vertices an edge connects
  • Adjacent vertices: Vertices connected by an edge
  • Loop: An edge with both end vertices identical
  • Parallel (multiple) edges: Two or more edges connecting the same pair of vertices
  • Degree (valency) of a vertex: The number of edges incident to that vertex (Loops contribute 2 to the degree)
  • Isolated vertex: A vertex with degree 0
  • Pendent vertex: A vertex with degree 1
  • Simple graph: A graph with no loops or parallel edges
  • Multigraph: A graph with loops or parallel edges

Handshaking Lemma

  • In any graph, the sum of the degrees of all vertices is equal to twice the number of edges
  • Σd(v) = 2m (where m is the number of edges)

Odd and Even Vertices

  • Odd vertex: A vertex with an odd degree
  • Even vertex: A vertex with an even degree
  • In any graph, the number of odd vertices is even

Special Types of Graphs

  • Null graph (Nₙ): A graph with no edges
  • Complete graph (Kₙ): A graph where every pair of vertices is connected by an edge
  • Note that K₁: 1 vertex, K₂: 2 vertices, K3.3: 2 sets of vertices with every vertex in one set connected to every vertex in the other set

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Applied Mathematics - II: Graph Theory PDF

Description

Test your knowledge of fundamental concepts in graph theory. This quiz covers critical definitions and properties, such as edges, vertices, loops, and the Handshaking Lemma. Challenge yourself to understand the distinctions between different types of graphs and their characteristics.

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