Let n ≥ 3. Show that if G is a graph with the same chromatic
polynomial...
Let n ≥ 3. Show that if G is a graph with the same chromatic
polynomial as Cn, then G is isomorphic to Cn. (Hint: Use induction.
What kind of graph must G − e be for any edge e? Why?)
Let G be a simple planar graph with no triangles.
(a) Show that G has a vertex of degree at most 3. (The proof was
sketched in the lectures, but you must write all the details, and
you may not just quote the result.)
(b) Use this to prove, by induction on the number of vertices,
that G is 4-colourable.
Let Q:= {1,2,...,q}. Let G be a graph with the elements of Q^n
as vertices and an edge between (a1,a2,...,an) and (b1,b2,...bn) if
and only if ai is not equal to bi for exactly one value of i. Show
that G is Hamiltonian.
Let G be a simple undirected graph with n vertices where n is an
even number. Prove that G contains a triangle if it has at least
(n^2 / 4) + 1 edges using mathematical induction.
let
G be a simple graph. show that the relation R on the set of
vertices of G such that URV if and only if there is an edge
associated with (u,v) is a symmetric irreflexive relation on
G
A maximal plane graph is a plane graph G = (V, E) with n ≥ 3
vertices such that if we join any two non-adjacent vertices in G,
we obtain a non-plane graph.
a) Draw a maximal plane graphs on six vertices.
b) Show that a maximal plane graph on n points has 3n − 6 edges
and 2n − 4 faces.
c) A triangulation of an n-gon is a plane graph whose infinite
face boundary is a convex n-gon...
Let N be a normal subgroup of the group G.
(a) Show that every inner automorphism of G defines an
automorphism of N.
(b) Give an example of a group G with a normal subgroup N and an
automorphism of N that is not defined by an inner automorphism of
G
Let P(x) be a polynomial of degree n
and A = [an , an-1,.... ]
Write a function integral(A, X1, X2) that takes 3 inputs A, X0
and X1
A as stated above
X1 and X2 be any real number, where X1 is the lower limit of the
integral and X2 is the upper limit of the integral. Please write
this code in Python.