Let $G$ be a graph. The line graph of $G$ is a new graph $L(G)$ whose vertices are the edges of $G$; two vertices of $L(G)$ are adjacent if, as edges of $G$, they share a common end point. In symbols:

$$V[L(G)]=E(G)\text{ and }E[L(G)]=\{e_1e_2:|e_1\cap e_2|=2\}$$

Prove or disprove the following statements about the relationship between a graph and its line graph : a. If $G$ is Eulerian, then $L(G)$ is also Eulerian. b. If $G$ has a Hamiltonian cycle, then $L(G)$ is Eulerian. (See Exercise 50.16 for the definition of a Hamiltonian cycle.) c. If $L(G)$ is Eulerian, then $G$ is also Eulerian. d. If $L(G)$ is Eulerian, then $G$ has a Hamiltonian cycle.

Name two different uses for the C++ operator *.

The __________ operator can be used to determine a variable’s address.

Briefly describe how the && operator works.

Determine whether the statement is true or false, and justify your answer. If there is a nonzero vector in the kernel of the matrix operator

$$T_A: R^n→R^n$$

then this operator is not one-to-one.

Consider a mass m constrained to move in a vertical line under the influence of gravity. Using the coordinate x measured vertically down from a convenient origin O, write down the Lagrangian L and find the generalized momentum p = ∂xL/∂ẋ. Find the Hamiltonian H as a function of x and p, and write down Hamilton's equations of motion. It is too much to hope with a system this simple that you would learn anything new by using the Hamiltonian approach, but do check that the equations of motion make sense.)

Briefly describe how the || operator works.

A ferry operator takes tourists to an island. The operator carries an average of 500 people per day for a round-trip fare of $20. The operator estimates that for each$1 increase in fare, 20 fewer people will take the trip. What fare will maximize the number of people taking the ferry?

$$\begin{array} { l } { \text { Find a Hamiltonian circuit in } } \\ { \text { Cay } \left( \{ ( a , 0 ) , ( b , 0 ) , ( e , 1 ) \} : Q _ { 4 } \oplus Z _ { m } \right) } \\ { \text { where } m \text { is even. } } \end{array}$$

The __________ operator is used to dynamically allocate memory.

Using the fact that the radial momentum operator is given by $\hat{p}_r=-i \hbar \frac{1}{r} \frac{\partial}{\partial r} r$, calculate the commutator $\left[\hat{r}, \hat{p}_r\right]$ between the position operator, $\hat{r}$, and the radial momentum operator.

Briefly describe how the or operator works

The Boolean operator ⊕, called the XOR operator, is defined by 1 ⊕ 1 = 0, 1 ⊕ 0 = 1, 0 ⊕ 1 = 1, and 0 ⊕ 0 = 0. Show that x ⊕ y = y ⊕ x.

Show that the operator $\hat{p}^{\prime}$ obtained by applying a translation to the operator $\hat{p}$ is $\hat{p}^{\prime}=\hat{T}^{\dagger} \hat{p} \hat{T}=\hat{p}$.