Definition: G has radius at most r if there is a vertex v such that every vertex in G can be reached from v in at most r steps. The radius of G is the smallest possible radius of any vertex, known as the center. Example: Radius of Pn is 4 with center being a middle vertex. Lemma: diameter < 2*radius Pf: any two vertices are reachable from each other by walking through the center. n(n-1) Lemma: The order of Kn is (") 2 Pf: each vertex is incident on n-1 edges, so n(n– 1) incidences overall. Since each edge has two vertices, we need to divide by two. Definition: the degree of a vertex v, degc(v) is the number of edges it is incident on. Corollary If every vertex has degree at least |V(G)|/2, then the graph is Hamiltonian. Proof: In that case deg(u) + deg(v) 2÷+ n = n for any two vertices, so Ore's 2 theorem applies.
Definition: G has radius at most r if there is a vertex v such that every vertex in G can be reached from v in at most r steps. The radius of G is the smallest possible radius of any vertex, known as the center. Example: Radius of Pn is 4 with center being a middle vertex. Lemma: diameter < 2*radius Pf: any two vertices are reachable from each other by walking through the center. n(n-1) Lemma: The order of Kn is (") 2 Pf: each vertex is incident on n-1 edges, so n(n– 1) incidences overall. Since each edge has two vertices, we need to divide by two. Definition: the degree of a vertex v, degc(v) is the number of edges it is incident on. Corollary If every vertex has degree at least |V(G)|/2, then the graph is Hamiltonian. Proof: In that case deg(u) + deg(v) 2÷+ n = n for any two vertices, so Ore's 2 theorem applies.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please help to understand thoroughly the concepts in the image attached
![Definition: G has radius at most r if there is a vertex v such
that every vertex in G can be reached from v in at most r steps.
The radius of G is the smallest possible radius of any vertex,
known as the center.
Example: Radius of P is 4
with center being a middle vertex.
Lemma: diameter < 2*radius
Pf: any two vertices are reachable from each other by walking
through the center.
п(п-1)
Lemma: The order of Kn is (“)
%3D
2
Pf: each vertex is incident on n-1 edges, so n(n– 1) incidences
overall. Since each edge has two vertices, we need to divide by
two.
Definition: the degree of a vertex v, degc(v) is the number of
edges it is incident on.
Corollary
If every vertex has degree at least
|V(G)|/2, then the graph is Hamiltonian.
п
Proof: In that case deg(u) + deg(v) >÷+
п
= n for any two vertices,
so Ore's
theorem applies.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F453ee218-b4a7-4772-b2c1-1b324abbebac%2F93d0dde3-e4ed-43c9-b994-001cd3643333%2Floepiqs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Definition: G has radius at most r if there is a vertex v such
that every vertex in G can be reached from v in at most r steps.
The radius of G is the smallest possible radius of any vertex,
known as the center.
Example: Radius of P is 4
with center being a middle vertex.
Lemma: diameter < 2*radius
Pf: any two vertices are reachable from each other by walking
through the center.
п(п-1)
Lemma: The order of Kn is (“)
%3D
2
Pf: each vertex is incident on n-1 edges, so n(n– 1) incidences
overall. Since each edge has two vertices, we need to divide by
two.
Definition: the degree of a vertex v, degc(v) is the number of
edges it is incident on.
Corollary
If every vertex has degree at least
|V(G)|/2, then the graph is Hamiltonian.
п
Proof: In that case deg(u) + deg(v) >÷+
п
= n for any two vertices,
so Ore's
theorem applies.
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