Question

Give an example of a graph (with or without weights on edges) where the betweenness and closeness centrality points are different. The graph must be composed of at least 5 vertices and at most 8 vertices.

Answer #1

**Answer:**

betweenness and closeness centrality points are different. The graph must be composed of at least 5 vertices and at most 8 vertices.

A graph is called planar if it can be
drawn in the plane without any edges crossing. The Euler’s formula
states that v − e + r = 2,
where v,e, and r are the numbers of vertices,
edges, and regions in a planar graph, respectively. For the
following problems, let G be a planar simple graph with 8
vertices.
Find the maximum number of edges in G.
Find the maximum number of edges in G, if G
has no...

Suppose that a connected graph without loops or parallel edges
has 11 vertices, each of degree 6. a. Must the graph have an Euler
Circuit? Explain b. Must the graph have a Hamilton Circuit? Explain
c. If the graph does have an Euler Circuit, how many edges does the
circuit contain? d. If the graph does have a Hamilton Circuit, what
is its length?

You are a mail deliverer. Consider a graph where the the streets
are the edges and the intersections are the vertices. You want to
deliver mail along each street exactly once without repeating any
edges. Would this path be represented by an Euler circuit or a
Hamiltonian circuit?
Write EUL for Euler circuit or
HAM for Hamiltonian circuit.
ANSWER:

Draw an example of a connected bipartite simple graph with 9
vertices and 10 edges that has an Euler tour.

Prove that if a graph has 1000 vertices and 4000 edges then it
must have a cycle of length at most 20.

Given a tree with p vertices, how many edges must you add
without adding vertices to obtain a maximal planar graph?

Give an example of a directed graph with capacities, and two
vertices s, t, where there are 2 distinct cuts which are both
minimum.

Consider edges that must be in every spanning tree of a graph.
Must every graph have such an edge? Give an example of a graph that
has exactly one such edge.

Give an example of a connected undirected graph that contains at
least twelve vertices that contains at least two circuits. Draw
that graph labeling the vertices with letters of the alphabet.
Determine one spanning tree of that graph and draw it. Determine
whether the graph has an Euler circuit. If so, specify the circuit
by enumerating the vertices involved. Determine whether the graph
has an Hamiltonian circuit. If so, specify the circuit by
enumerating the vertices involved.

Let G be a connected simple graph with n vertices and m edges.
Prove that G contains at least m−n+ 1 different subgraphs
which are polygons (=circuits). Note: Different polygons
can have edges in common. For instance, a square with a diagonal
edge has three different polygons (the square and two different
triangles) even though every pair of polygons have at least one
edge in common.

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