One of the simplest algorithms for determining connectivity is breadth$-$first search $($BFS$).$ It explores outward from source, adding nodes one $$layer$$ at a time. Initially, it includes nodes connected to source by an edge $($the first layer$).$ Then, it adds nodes connected to any node in the first layer $($the second layer$),$ and so on until no new nodes are encountered. Answer the following questions:

a$)$ Let T be a BFS tree generated from graph G$,$ let x and y be nodes in T belonging to layers L$\_$i and L$\_$j respectively, and let $($x$,$ y$)$ be an edge of G$.$ What is the maximum and minimum difference between i and j $?$

b$)$ For any j $>=1,$ what is the maximum and minimum distance of any nodes residing at layer L$\_$j from the source node $?$