Which one of the following is the closed form for the generating function of the sequence (a_{n}}_{n $$\ge$$ 0} defined below?

$${a_n} = \left\{ {\matrix{ {n + 1,} & {n\,is\,odd} \cr {1,} & {otherwise} \cr } } \right.$$

Consider a simple undirected unweighted graph with at least three vertices. If A is the adjacency matrix of the graph, then the number of 3-cycles in the graph is given by the trace of

Consider solving the following system of simultaneous equations using LU decomposition.

x_{1} + x_{2} $$-$$ 2x_{3} = 4

x_{1} + 3x_{2} $$-$$ x_{3} = 7

2x_{1} + x_{2} $$-$$ 5x_{3} = 7

where L and U are denoted as

$$L = \left( {\matrix{ {{L_{11}}} & 0 & 0 \cr {{L_{21}}} & {{L_{22}}} & 0 \cr {{L_{31}}} & {{L_{32}}} & {{L_{33}}} \cr } } \right),\,U = \left( {\matrix{ {{U_{11}}} & {{U_{12}}} & {{U_{13}}} \cr 0 & {{U_{22}}} & {{U_{23}}} \cr 0 & 0 & {{U_{33}}} \cr } } \right)$$

Which one of the following is the correct combination of values for L_{32}, U_{32}, and x_{1} ?

Consider a simple undirected weighted graph G, all of whose edge weights are distinct. Which of the following statements about the minimum spanning trees of G is/are TRUE?