If G = (V, E) is a graph and x ∈ V , let G \ x be the graph
whose vertex set is V \ {x} and whose edges are those edges of G
that don’t contain x.
Show that every connected finite graph G = (V, E) with at least
two vertices has at least two vertices x1, x2 ∈ V such that G \ xi
is connected.
Let Z2 [x] be the ring of all polynomials with coefficients in Z2. List the elements of the field Z2 [x]/〈x2+x+1〉, and make an addition and multiplication table for the field. For simplicity, denote the coset f(x)+〈x2+x+1〉 by (f(x)) ̅.
Let G be a connected graph and let e be a cut edge in G.
Let K be the subgraph of G defined by:
V(K) = V(G) and
E(K) = E(G) - {e}
Prove that K has exactly two connected components. First
prove that e cannot be a loop. Thus the endpoint set of e is of the
form {v,w}, where v ≠ w. If ṽ∈V(K), prove that either there is a
path in K from v to ṽ, or...
Q1-Find all possible time domain signals corresponding
to the following z-transform:
X(z) = (z3 + z2 + 3/2 z + 1/2 ) /
(z3 + 3/2 z2 + 1/2 z)
Q2-A digital linear time invariant filter has the
following transfer function:
H(z) = (5 + 5z-1) / (1 - 3/8 z-1 + 1/16
z-2)
a) Find the impulse response if the filter is causal.