Question

Prove

i.) The line graph of P_{n} is P_{n-1}

ii.) The line graph of C_{n} is C_{n}

Answer #1

i) No. of edges in Pn is n. Hence L(Pn) has n-1 vertices.Name them 1,2,..,n-1. edges i and i+1 have common point hence in L(Pn) vertices i and i+1 are adjacent and so it is isomorphic to P(n-1)

ii) Consider a cycle with n vertices, it has n edges in its path. By these edges as vertices in L(Cn), we form a cycle with n edges. This implies that, both Cn and L(Cn) has same number of vertices and edges and also have the same degree sequence. So, there exists a 1-1 correspondence between V(Cn) into V(L(Cn)) and E(Cn) and E( L(Cn)) and preserves adjacency

. Prove that, for all integers n ≥ 1, Pn i=1 i(i!) = (n + 1)! −
1

Exercise 3. Let Wn be the graph obtained from the cycle graph Cn
by adding one new vertex which is adjacent to every vertex of Cn.
Prove that for n ≥ 3, Wn does not have an Eulerian trail.

LED (PN junctions): if you have a graph of V (x-axis)
vs ln(I) (y-axis), how do you calculate Vt (thermal voltage) and Is
(saturation current)?
While plotting the graph, do you take ln of the current in amps or
mA?

show that product graph K4 X Pn is graceful
labelling

Graph a monopolistically competitive firm in both (i) SR, and
(ii) LR.

(i)Prove that if
∀a∈A ∃b∈B s.t. a≤b,
then supA≤supB.
(ii)Prove that if
∀a∈A ∃b∈B s.t. a≥b,
then infB≤infA

Potassium hexachloroferrate(II) can be dissolved in water and
reacted with cyanide ion (CN-) to form the more favorable
cyano-complex forms:
K4[FeCl6] + 6 CN- ===> K4[Fe(CN)6] + 6 Cl-
A student performs this experiment by adding 20 mL of 4.0 M
NaCN (aq) to 4.562 g of K4[FeCl6]. The solid product was isolated,
dried, and massed. Answer the following questions based on this
information.
1. How many moles of potassium hexachloroferrate(II) are
available at the start?
2. How many moles...

Graph G is a connected planar graph with 1 face. If G is finite,
prove that there is a vertex with degree 1.

Consider a population that grows according to the
recursive rule Pn=Pn−1+90Pn=Pn-1+90, with
initial population P0=50
P0=50.
Then:
P1 =
P2 =
Find an explicit formula for the population. Your formula
should involve nn (use lowercase n)
Pn=
Pn=
Use your explicit formula to find P100
P100 =

Construct a type-II pn heterojunction and schematically draw its
band diagrams in both charge neutral and equilibrium conditions

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