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Questions and Answers
For definiteness, say G₁ and G₂ are the two ______.
For definiteness, say G₁ and G₂ are the two ______.
components
The ______ Seven Bridge Problem is a classic problem in graph theory.
The ______ Seven Bridge Problem is a classic problem in graph theory.
Konigsberg
The problem was to find a way to walk through all seven bridges without crossing any ______ more than once.
The problem was to find a way to walk through all seven bridges without crossing any ______ more than once.
bridge
A graph G is ______ if and only if all circuits (cycles) in G are of even length.
A graph G is ______ if and only if all circuits (cycles) in G are of even length.
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Let V₁ and V₂ be the two sets of vertices in the ______, with every edge connecting one vertex from V₁ to a vertex from V₂.
Let V₁ and V₂ be the two sets of vertices in the ______, with every edge connecting one vertex from V₁ to a vertex from V₂.
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If G is a ______ graph on n vertices, then n is of the type 4k or 4k + 1 for some integer k.
If G is a ______ graph on n vertices, then n is of the type 4k or 4k + 1 for some integer k.
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A ______ graph G is a graph where, for every pair of vertices, there is at least one path connecting them.
A ______ graph G is a graph where, for every pair of vertices, there is at least one path connecting them.
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The number of ______ components in graph is represented as W(G).
The number of ______ components in graph is represented as W(G).
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A graph G = (V, E) consists of a set V whose elements are called ______.
A graph G = (V, E) consists of a set V whose elements are called ______.
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An edge in a graph connects two vertices and is associated with an unordered pair, known as the ______ of the edge.
An edge in a graph connects two vertices and is associated with an unordered pair, known as the ______ of the edge.
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An edge whose end vertices are identical is called a ______.
An edge whose end vertices are identical is called a ______.
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If two edges share the same end vertices, they are referred to as ______ edges.
If two edges share the same end vertices, they are referred to as ______ edges.
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A vertex with a degree of zero is called an ______.
A vertex with a degree of zero is called an ______.
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The ______ is a principle stating that the sum of the degrees of all vertices in a graph equals twice the number of edges.
The ______ is a principle stating that the sum of the degrees of all vertices in a graph equals twice the number of edges.
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A graph without loops and parallel edges is termed a ______.
A graph without loops and parallel edges is termed a ______.
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In any graph, the number of odd vertices is ______.
In any graph, the number of odd vertices is ______.
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A subgraph obtained by deleting a vertex and all edges incident on it is called a ______ deleted subgraph.
A subgraph obtained by deleting a vertex and all edges incident on it is called a ______ deleted subgraph.
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If U is a non-empty subset of the vertex set V, the subgraph G<U> is a ______ induced subgraph.
If U is a non-empty subset of the vertex set V, the subgraph G<U> is a ______ induced subgraph.
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If F is a non-empty subset of the edge set E, the subgraph G<F> is an ______ induced subgraph.
If F is a non-empty subset of the edge set E, the subgraph G<F> is an ______ induced subgraph.
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A ______ is defined as a finite alternating sequence of vertices and edges.
A ______ is defined as a finite alternating sequence of vertices and edges.
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A ______ is a walk in which no edge is repeated.
A ______ is a walk in which no edge is repeated.
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A closed walk in which no vertex is repeated except terminal vertices is called a ______.
A closed walk in which no vertex is repeated except terminal vertices is called a ______.
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A graph is called ______ if every two of its vertices are connected.
A graph is called ______ if every two of its vertices are connected.
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A graph G = (V, E) is called a null graph if the edge set E is ______.
A graph G = (V, E) is called a null graph if the edge set E is ______.
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If every vertex of graph G is of the same degree K, we say that G is ______-regular.
If every vertex of graph G is of the same degree K, we say that G is ______-regular.
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A spanning subgraph of G is a subgraph H with V(H) = ______(G).
A spanning subgraph of G is a subgraph H with V(H) = ______(G).
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Removing a single ______ will cause a disconnection in the network represented by graph G.
Removing a single ______ will cause a disconnection in the network represented by graph G.
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Let G be a graph. If the vertex set V of G can be partitioned into two non-empty subsets V₁ and V₂, then G is called a ______ Graph.
Let G be a graph. If the vertex set V of G can be partitioned into two non-empty subsets V₁ and V₂, then G is called a ______ Graph.
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A graph H = (V(H), E(H)) is called a subgraph of a graph G = (V(G), E(G)) if V(H) ⊆ ______(G) and E(H) ⊆ E(G).
A graph H = (V(H), E(H)) is called a subgraph of a graph G = (V(G), E(G)) if V(H) ⊆ ______(G) and E(H) ⊆ E(G).
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A proper subgraph H of G is defined as H ⊆ G but H ______ G.
A proper subgraph H of G is defined as H ⊆ G but H ______ G.
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In a complete graph, there is an edge between ______ pair of vertices.
In a complete graph, there is an edge between ______ pair of vertices.
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A simple graph G is not connected if there exists two vertices u and v in G such that there is no edge joining u and v. This means G is a _____ union of its subgraphs.
A simple graph G is not connected if there exists two vertices u and v in G such that there is no edge joining u and v. This means G is a _____ union of its subgraphs.
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The complement graph G contains edges that are not present in G, which means for any edge e = (u, v), e ∉ E(G) implies e ∈ _____ G.
The complement graph G contains edges that are not present in G, which means for any edge e = (u, v), e ∉ E(G) implies e ∈ _____ G.
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According to Theorem 3, a graph G is connected if for every partition of its vertex set V(G) into two non-empty subsets V₁ and V₂, there exists an edge of G with one vertex in V₁ and another in _____
According to Theorem 3, a graph G is connected if for every partition of its vertex set V(G) into two non-empty subsets V₁ and V₂, there exists an edge of G with one vertex in V₁ and another in _____
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If a graph has exactly two vertices of odd degree, then there must be a _____ between these two vertices.
If a graph has exactly two vertices of odd degree, then there must be a _____ between these two vertices.
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In the base case of the proof for Theorem 4, when m = 0, there are no edges, so the graph has n _____
In the base case of the proof for Theorem 4, when m = 0, there are no edges, so the graph has n _____
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If G is disconnected, it can be expressed as a disjoint union of subgraphs, such as G equals G₁ union G₂. There does not exist an edge joining the vertex of G₁ and the vertex of _____
If G is disconnected, it can be expressed as a disjoint union of subgraphs, such as G equals G₁ union G₂. There does not exist an edge joining the vertex of G₁ and the vertex of _____
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The proof of Theorem 5 includes two cases: the first case considers G as being _____, which allows for a direct path between vertices.
The proof of Theorem 5 includes two cases: the first case considers G as being _____, which allows for a direct path between vertices.
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Theorem 4 indicates that for a simple graph with n vertices, m edges, and k components, the inequality states n - k ≤ _____.
Theorem 4 indicates that for a simple graph with n vertices, m edges, and k components, the inequality states n - k ≤ _____.
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Study Notes
Graph Theory
- Graph theory differs from function graphs.
- Informally, a graph consists of vertices and edges.
- Vertices represent entities (e.g., people).
- Edges connect vertices indicating a relationship.
- Example applications include acquaintance, chemical bonding, electrical networks, and transportation networks.
- Graph theory uses various techniques, including induction, parity, extremality, counting (in two ways), Pigeonhole Principle, and Inclusion-Exclusion principles.
Types of Graphs
- Null graph (Nn): A graph with no edges.
- Complete graph (Kn): A simple graph where each pair of distinct vertices is connected by an edge.
- Regular graph: A graph where all vertices have the same degree.
- Bipartite graph: A graph whose vertices can be divided into two disjoint sets such that every edge connects a vertex in one set to a vertex in the other set.
Elementary Terminologies and Results
- End vertices: The vertices an edge connects.
- Adjacent vertices: Vertices connected by an edge.
- Loop: An edge whose both end nodes are the same.
- Parallel edges: Two or more edges connecting the same pair of vertices.
- Degree (or valency) of a vertex: The number of edges incident on the vertex. A loop contributes 2 to the degree of the vertex.
- Isolated vertex: A vertex with a degree of zero.
- Pendent vertex: A vertex with a degree of one.
- Simple graph: A graph with no loops and no parallel edges.
- Multigraph: A graph that can have parallel edges or loops.
- (n, m) graph: A graph with n vertices and m edges.
Theorem 1 (Handshaking Lemma)
- The sum of the degrees of all vertices in a graph equals twice the number of edges.
Special Graphs
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Null graph Nn: Graph with n vertices and no edges.
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Complementary graph: G' is the complementary graph of G, where there is an edge between two vertices in G' if there is no corresponding edge in G.
Connected Graphs
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Connected graph: A graph where there is a path between every pair of vertices.
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Disconnected graph: A graph that contains at least two disconnected components
Walk, Trail, and Cycle in a Graph
- Walk: A sequence of vertices and edges where each edge connects consecutive vertices.
- Trail: A walk in which no edge is repeated. Repetition of vertices is allowed.
- Path: A trail where no vertex is repeated.
- Cycle: A closed walk or trail that starts and ends at the same vertex.
Induced Subgraphs
- Induced Subgraph (G[U]): Subgraph of G where only the vertices in U and edges determined by them are included.
- Vertex deletion: (G -v): Subgraph of G after removing vertex v and all incident edges.
- Edge deletion: (G -e) Subgraph of G after removing edge e and preserving all other vertices.
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Description
Test your knowledge on graph theory concepts and terminology with this engaging quiz. Topics include the Seven Bridge Problem, types of graphs, vertices, edges, and components within graphs. Challenge yourself and deepen your understanding of this fundamental area in mathematics.
