Question

Consider the fully connected graph K_{2020.}. What is
the maximum number of edges which may be removed while preserving
the maximum degree of K_{2020}?

Answer #1

The distance between two connected nodes in a graph is the
length (number of edges) of the shortest path connecting them. The
diameter of a connected graph is the maximum distance between any
two of its nodes. Let v be an arbitrary vertex in a graph G. If
every vertex is within distance d of v, then show that the diameter
of the graph is at most 2d.

Exercise 29 .
What is the largest possible number of vertices in a connected
graph with 35 edges, all vertices having degree at least 3? Can you
verify your result and ﬁnd a graph with such properties?

Let there be planar graph G with 12 vertices where every
vertices may or may not be connected by an edge. The edges in G
cannot intersect. What is the maximum number of edges in G. Draw an
example of G. What do you notice about the faces and the maximum
number of edges?

Discrete Math:
What is the maximum number of edges on a simple disconnected
graph with n vertices. Justify your answer. Please write clearly
and do not skip steps. Thank you

Call a graph on n vertices dendroid if it has n edges and is
connected. Characterize degree sequences of dendroids.

Find two simple connected graphs with the same number of edges
and the same number of vertices which are not isomorphic.
Please draw solutions.
(Graph Theory)

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?

what is the fewest number of edges a graph could have in terms
of the number of vertices?

Consider the complete bipartite graph Kn,n with 2n vertices. Let
kn be the number of edges in Kn,n. Draw K1,1, K2,2 and K3,3 and
determine k1, k2, k3. Give a recurrence relation for kn and solve
it using an initial value.

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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