Let $$G$$ $$\,\,\,\,\, = \,\,\,\left( {V,\,\,\,\,\,E} \right)$$ be a graph. Define $$\xi \left( G \right) = \sum\limits_d {{i_d} \times } {\mkern 1mu} d,$$ where $${{i_d}}$$ is the number of vertices of degree $$d$$ in $$G$$. If $$S$$ and $$T$$ are two different trees with $$\xi \left( S \right) = \xi \left( T \right)$$, then

What is the chromatic number of an $$n$$-vertex simple connected graph which does not contain any odd length cycle? Assume $$n \ge 2$$.

Which one of the following is TRUE for any simple connected undirected graph with more than $$2$$ vertices?

What is the size of the smallest MIS (Maximal Independent Set) of a chain of nine nodes?