Lecture 10: Graphs
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Questions and Answers

What term describes the number of edges connecting to a vertex?

  • Degree (correct)
  • Cycle
  • Path
  • Circuit

Which of the following describes a closed path that revisits no vertices?

  • Path
  • Cycle (correct)
  • Circuit
  • Simple Path

If removing an edge increases the number of components in a graph, what is this edge called?

  • Cycle
  • Bridge (correct)
  • Circuit
  • Path

What type of graph is characterized by the presence of a path between any two of its vertices?

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

What is the term for a sequence of vertices where each vertex is adjacent to the next?

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

Which of the following best describes adjacent vertices?

<p>Vertices connected by an edge (B)</p> Signup and view all the answers

What defines an Euler path in a graph?

<p>Travels through all edges (C)</p> Signup and view all the answers

What is a loop in the context of graph theory?

<p>An edge that connects a vertex to itself (D)</p> Signup and view all the answers

What defines an Euler circuit?

<p>A circuit that visits all edges of a connected graph (A)</p> Signup and view all the answers

Which type of graph does NOT have any edges?

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

What is a characteristic of a simple graph?

<p>It has no parallel edges and no loops (D)</p> Signup and view all the answers

Which of the following describes a directed graph?

<p>Edges are directed by arrows (C)</p> Signup and view all the answers

Which of the following is an example of how graphs can represent relationships?

<p>Transport networks showing town connections (D)</p> Signup and view all the answers

What is true about a bipartite graph?

<p>It consists of two disjoint sets of vertices (D)</p> Signup and view all the answers

Which statement is correct regarding cyclic and acyclic graphs?

<p>Cyclic graphs have edges that form cycles, acyclic do not (B)</p> Signup and view all the answers

How are adjacent vertices defined in a graph?

<p>Vertices that share an edge (D)</p> Signup and view all the answers

What characterizes a complete graph?

<p>There is exactly one edge connecting every pair of vertices. (D)</p> Signup and view all the answers

Which description applies to a disconnected graph?

<p>It includes vertices that cannot reach others due to independent components. (A)</p> Signup and view all the answers

What defines a regular graph?

<p>All vertices have the same degree. (B)</p> Signup and view all the answers

What is a cyclic graph?

<p>A graph with at least one cycle. (B)</p> Signup and view all the answers

Which of the following is true for an acyclic graph?

<p>It does not contain any cycle. (B)</p> Signup and view all the answers

What is the main feature of a bipartite graph?

<p>It has edges that only connect vertices from two distinct sets. (A)</p> Signup and view all the answers

In a connected graph, what is guaranteed?

<p>It is possible to visit one vertex from any other vertex. (C)</p> Signup and view all the answers

What condition must a graph meet to be classified as k-regular?

<p>Every vertex must be connected to k other vertices. (A)</p> Signup and view all the answers

What is the eccentricity of vertex 'd'?

<p>2 (D)</p> Signup and view all the answers

Which vertex serves as the central point of the graph?

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

What is the value of the diameter of the graph, denoted as d(G)?

<p>3 (C)</p> Signup and view all the answers

How is the radius of a connected graph defined?

<p>Minimum eccentricity from all vertices (C)</p> Signup and view all the answers

What is the circumference of the graph?

<p>6 (B)</p> Signup and view all the answers

Which of the following vertices has the highest eccentricity?

<p>a (A), b (B), f (C), e (D)</p> Signup and view all the answers

Given the distances from vertex 'a', which pair yields the eccentricity of 3?

<p>a to g (D)</p> Signup and view all the answers

If the eccentricity of the graph equals its radius, what does this indicate?

<p>The graph's maximum path is minimal (A)</p> Signup and view all the answers

What does an adjacency matrix indicate about the vertices in a graph?

<p>It shows which vertices are connected by edges. (A)</p> Signup and view all the answers

In an incidence matrix, what does a value of -1 signify?

<p>The edge is connected as an incoming edge to the vertex. (A)</p> Signup and view all the answers

Which graph representation uses a linked structure to maintain lists of adjacent vertices?

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

What is the size of the incidence matrix for a graph with 5 vertices and 3 edges?

<p>5x3 (B)</p> Signup and view all the answers

How is an edge indicated in an adjacency matrix for vertices i and j?

<p>ai,j = 1 if there is an edge. (A)</p> Signup and view all the answers

Which of the following representations is optimal for a sparsely connected graph?

<p>Adjacency list (D)</p> Signup and view all the answers

For a weighted graph, what additional information must be associated with each edge?

<p>The weight of the edge. (C)</p> Signup and view all the answers

What is a key characteristic of both directed and undirected graphs in representation?

<p>They can both be represented using incidence matrices. (A)</p> Signup and view all the answers

What characterizes a complete bipartite graph?

<p>Every pair of vertices in the sets are connected by exactly one edge. (B)</p> Signup and view all the answers

How is a star graph represented and defined?

<p>It is characterized by one central vertex connected to n-1 outer vertices, all with degree 1. (B)</p> Signup and view all the answers

What defines the length of a path in a weighted graph?

<p>The sum of the weights of all the edges included in the path. (C)</p> Signup and view all the answers

Which statement is true about multi-graphs?

<p>There can be multiple edges connecting the same pair of vertices. (C)</p> Signup and view all the answers

What distinguishes a planar graph from a non-planar graph?

<p>Planar graphs can be drawn without crossing edges, while non-planar graphs cannot. (D)</p> Signup and view all the answers

How is the distance between two vertices defined?

<p>As the number of edges in the shortest path connecting the vertices. (A)</p> Signup and view all the answers

What does the eccentricity of a vertex represent?

<p>The maximum distance from that vertex to all other vertices. (C)</p> Signup and view all the answers

In a multi-graph, what is true if vertices B and C are connected by two edges?

<p>This graph cannot be a complete bipartite graph. (D)</p> Signup and view all the answers

Flashcards

What is a graph?

A pictorial and mathematical representation of a set of objects where some pairs of objects are connected by links.

What are nodes in a graph?

The interconnected objects in a graph are called vertices or nodes.

What are edges in a graph?

The links that connect the vertices in a graph are called edges, arcs, or lines.

What are neighbours in a graph?

When two nodes are connected by an edge, they are called neighbours.

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What is the degree of a node?

The degree of a node is the number of other nodes that it is connected to.

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What is a loop in a graph?

A loop is an edge that connects a node to itself.

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What is a path in a graph?

A path is a sequence of nodes connected by edges.

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What is a cycle in a graph?

A cycle is a closed path that starts and ends at the same node, and no node is visited more than once.

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Null Graph

A graph with no edges between its vertices.

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Trivial Graph

A graph with only one vertex, no edges.

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Simple Graph

A graph with undirected edges, no parallel edges or loops.

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Directed Graph

A graph with edges that have a direction.

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Undirected Graph

A graph where all edges are undirected.

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Euler Circuit

A circuit that visits every edge exactly once in a connected graph.

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Graph

A graph that represents complex relationships between objects using nodes (vertices) connected by edges (arcs).

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Adjacency in a Graph

If there's an edge (u,v) in a graph, we say that node u is adjacent to node v.

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What is a complete graph?

A graph where all vertices are connected by exactly one edge.

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What is a connected graph?

A graph where there's a path between every pair of vertices.

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What is a disconnected graph?

A graph where no path exists between some pairs of vertices.

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What is a regular graph?

A graph where all vertices have the same degree, meaning they are connected to the same number of other vertices.

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What is a cyclic graph?

A graph with 'n' vertices and 'n' edges forming a cycle of 'n' with all its edges.

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What is an acyclic graph?

A graph that does not contain any cycles.

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What is a bipartite graph?

A graph whose vertices can be split into two sets, with edges only connecting vertices from different sets.

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What is a bipartite graph?

A graph whose vertices can be split into two sets, with edges only connecting vertices from different sets. Each edge connects one vertex from each set.

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Eccentricity of a vertex

The maximum distance from a vertex to any other vertex in a graph.

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Radius of a graph (r(G))

The minimum eccentricity among all vertices in a connected graph.

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Diameter of a graph (d(G))

The maximum eccentricity among all vertices in a graph.

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Central point of a graph

A vertex with eccentricity equal to the radius of the graph.

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Circumference of a graph

The number of edges in the longest cycle of a graph.

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Graph Representation

A method of representing a graph in a computer's memory.

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Complete Bipartite Graph

A bipartite graph where every vertex in the first set is connected to every vertex in the second set by a single edge.

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Star Graph

A specific kind of complete bipartite graph where one vertex (the 'center') has a degree equal to (n-1), while all other vertices have degree 1.

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Weighted Graph

A graph where each edge has a numerical weight assigned to it. The total weight of a path is the sum of the weights of its edges.

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Multi-graph

A graph that allows multiple edges connecting the same vertices, or edges connecting a vertex to itself (loops).

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Planar Graph

A graph that can be drawn on a flat surface without any edge crossing another edge (except at shared vertices).

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Non-Planar Graph

A graph that cannot be drawn on a flat surface without at least one pair of its edges crossing.

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Distance between two vertices

The shortest number of edges between two vertices in a graph. If there are multiple paths, we use the shortest one.

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Adjacency Matrix

A matrix representation of a graph where each row and column represents a vertex, and the elements indicate the presence or absence of an edge between vertices.

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Incidence Matrix

A representation of a graph using a matrix where each row represents a vertex and each column represents an edge. The entries in the matrix indicate whether a vertex is incident to an edge.

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Adjacency List

A representation of a graph where each vertex is associated with a list of its neighboring vertices. It's a linked representation.

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Directed Graph Representation (Adjacency Matrix)

A matrix where ai,j = 1 if there's an edge from vertex i to vertex j, and 0 otherwise, representing the presence or absence of a directed edge.

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Undirected Graph Representation (Adjacency Matrix)

A matrix where ai,j = 1 if there's an edge between vertices i and j, and 0 otherwise, showing undirected connections.

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Undirected Weighted Graph

Similar to an undirected graph, but with weights assigned to each edge, representing additional information about the connection.

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Directed Weighted Graph

Similar to an undirected graph, but with weights assigned to each edge, representing additional information about the connection.

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Study Notes

Lecture 10: Graphs

  • A graph is a pictorial and mathematical representation of objects where some pairs of objects are connected by links.

  • Interconnected objects are represented by points called vertices or nodes.

  • Links connecting vertices are called edges or arcs or lines.

  • A graph is an ordered pair G where:

    • G specifies the graph.
    • V is the vertex-set (elements are vertices or nodes); denoted by V(G) or V.
    • E is the edge-set (elements are edges or connections between vertices); denoted by E(G) or E.

  • Graphs have key terms:

    • Neighbors: Two nodes connected by an edge.
    • Degree of a node: The number of other nodes it is connected to (number of neighbors).
    • Loop: An edge connecting a node to itself.
    • Path: A sequence of nodes connected by edges.
    • Cycle: A closed path starting and ending at the same node (no node visited more than once).

  • Adjacent Vertices: Two vertices are adjacent if there's an edge connecting them.

  • Adjacent Edges: Edges sharing a common vertex.

  • Degree of a Vertex: The number of edges connected to that vertex.

  • Path: A sequence of vertices where each vertex is adjacent to the next. A simple path does not repeat vertices.

  • Circuit: A path that begins and ends at the same vertex.

  • Cycle: A circuit that doesn't repeat vertices.

  • Connected Graph: A graph where any two vertices are connected by a path.

  • Disconnected Graph: A graph that is not connected.

  • Components: Connected subgraphs in a disconnected graph.

  • Bridge: An edge whose deletion increases the number of components.

  • Euler Path: A path that travels through all edges of a connected graph.

  • Euler Circuit: A circuit that visits every edge of a connected graph.

  • Graph Types:

    • Null Graph: A graph with no edges between vertices (empty graph).
    • Trivial Graph: A graph with only one vertex.
    • Simple Graph: An undirected graph with no parallel edges and no loops.
    • Undirected Graph: Edges have no direction.
    • Directed Graph: Edges have direction (arrows).
    • Complete Graph: Every pair of vertices is connected by exactly one edge.
    • Connected Graph: Any two vertices are connected by a path.
    • Disconnected Graph: Not all vertices are connected.
    • Regular Graph: All vertices have the same degree. K-regular graph means all vertices have degree k.
    • Cyclic Graph: A graph with at least one cycle.
    • Acyclic Graph: A graph with no cycles.
    • Bipartite Graph: Vertices can be divided into two disjoint sets, and edges connect vertices in different sets only.
    • Complete Bipartite Graph: Every vertex in one set is connected to every vertex in the other set.
    • Star Graph: A complete bipartite graph where all vertices in one set have degree 1, and a single vertex connects to all of them.
    • Weighted Graph: Edges have weights/values assigned to them, allowing for the concept of path length.

  • Graph representations: Adjacency matrix, Incidence matrix, Adjacency list.

    • Adjacency Matrix: n x n matrix where aij = 1 if there is an edge between vertex i and j, otherwise 0.
    • Incidence Matrix: Rows for vertices, columns for edges. A non-zero represents connection with an edge.
    • Adjacency List: A linked list for each vertex, detailing adjacent vertices/edges.

  • Basic Properties:

    • Distance between two vertices: The number of edges in the shortest path between the vertices.
    • Eccentricity of a vertex: Maximum distance between a vertex and all other vertices.
    • Radius of a graph: Minimum eccentricity among all vertices.
    • Diameter of a graph: Maximum eccentricity among all vertices.
    • Central point(s): Vertex(vertices) with minimum eccentricity.
    • Circumference: The length of the longest cycle.

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Description

Explore the fascinating world of graphs in this quiz. Understand key concepts such as vertices, edges, paths, and cycles, as well as their applications in various fields. Test your knowledge on the properties and terminologies associated with graphs.

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