Question

Let *G* be a graph with vertex set *V*. Define a
relation *R* from *V* to itself as follows: vertex
*u* has this relation *R* with vertex *v*,
*u R v*, if there is a path in *G* from *u* to
*v*. Prove that this relation is an equivalence relation.
Write your proof with complete sentences line by line in a logical
order. If you can, you may write your answer to this
question directly in the space provided.**Your presentation
counts**.

Answer #1

Graph Theory
Let v be a vertex of a non trivial graph G. prove that if G is
connected, then v has a neighbor in every component of G-v.

let G be a connected graph such that the graph formed by
removing vertex x from G is disconnected for all but exactly 2
vertices of G. Prove that G must be a path.

Define on the vertices of a graph G u ≈ v if the distance d(u,
v) is even. Is this an equivalence relation? If you say yes than
show that it satisfies all properties. If you say no than show me
which ones are satisfied and which are not. Justify your
answers.

Let G be a graph whose vertex set is a set V = {p1,
p2, p3, . . . , p6} of six people.
Prove that there exist three people who are all friends with each
other, or three people none of whom are friends with each
other.

Let u and v be distinct vertices in a graph G. Prove that there
is a walk from ? to ? if and only if there is a path from ? to
?.

Let G be a graph or order n with independence number α(G) =
2.
(a) Prove that if G is disconnected, then G contains K⌈ n/2 ⌉ as
a subgraph.
(b) Prove that if G is connected, then G contains a path (u, v,
w) such that uw /∈ E(G) and every vertex in G − {u, v, w} is
adjacent to either u or w (or both).

You are given a directed acyclic graph G(V,E), where each vertex
v that has in-degree 0 has a value value(v) associated with it. For
every other vertex u in V, define Pred(u) to be the set of vertices
that have incoming edges to u. We now define value(u) = ?v∈P red(u)
value(v). Design an O(n + m) time algorithm to compute value(u) for
all vertices u where n denotes the number of vertices and m denotes
the number of edges...

Let G be a graph where every vertex has odd degree, and G has a
perfect matching. Prove that if M is a perfect matching of G, then
every bridge of G is in M.
The Proof for this question already on Chegg is wrong

Let p and q be any two distinct prime numbers and define the
relation a R b on integers a,b by: a R b iff b-a is divisible by
both p and q. For this relation R: Show that the equivalence
classes of R correspond to the elements
of ℤpq. That is: [a] = [b] as equivalence
classes of R if and only if [a] = [b] as elements of
ℤpq.
you may use the following lemma: If p is prime...

Let p and q be any two distinct prime numbers and define the
relation a R b on integers a,b by: a R b iff b-a is divisible by
both p and q. For this relation R: Prove that R is an equivalence
relation.
you may use the following lemma: If p is prime and p|mn, then
p|m or p|n

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 7 minutes ago

asked 22 minutes ago

asked 29 minutes ago

asked 35 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago