Question

Show that an undirected graph G = (N,A) is connected if and only
if for every partition of N into subsets N1 and N2, some arc has
one endpoint in N1 and the other endpoint in N2.

Answer #1

Let G be an undirected graph with n vertices and m edges. Use a
contradiction argument to prove that if m<n−1, then G is not
connected

Let G = (V, E) be an undirected and connected graph with
Laplacian matrix L.
(a) How are the eigenvalues of L2 related to the
eigenvalues of L?
(b) If instead of running the consensus protocol, x ̇ = −Lx, one
runs the protocol from Homework 1, given by
x ̇ = −L2x, will consensus still be achieved? Justify
your answer.
(c) Assuming both x ̇ = −Lx and x ̇ = −L2x converge,
which protocol converges faster? Justify your...

Show that a graph is connected if and only if there is no
bipartition of the set of its vertices such that no edge has an
endvertex in each subset of this bipartition.
Graph Theory

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 graph. Show that G has a subtree that is a
maximal tree.

Prove or disapprove each of the following:
(a) Every disconnected graph has an isolated vertex.
(b) A graph is connected if and only if some vertex is connected
to all other vertices.
(c) If G is a simple, connected, Eulerian graph, with edges e, f
that are incident to a common vertex, then G has an Eulerian
circuit in which e and f appear consequently.

Let G be a simple graph with n(G) > 2. Prove that G is
2-connected iff for every set of 3 distinct vertices, a,
b and c, there is an a,c-path
that contains b.

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.

Show that an edge e of a connected graph G belongs to any
spanning tree of G if and only if e is a bridge of G. Show that e
does not belong to any spanning tree if and only if e is a loop of
G.

A spanning tree of connected graph G = (V, E) is an acyclic
connected subgraph (V, E0 ) with the same vertices as G. Show that
every connected graph G = (V, E) contains a spanning tree. (It is
the connected subgraph (V, E0 ) with the smallest number of
edges.)

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