Let $G=(V,E)$ be a connected, undirected graph with real number edge weights. Also assume that all edge weights are distinct.

a$.(5$ pts$)$ Does the min$-$weight edge of $G$ have to be on Minimum Spanning Tree $($MST$)$ of $G?$

b$.(5$ pts$)$ Can the max$-$weight edge of $G$ belong to MST$?$

c$.(5$ pts$)$ Adding the same positive value to every edge of $G$ can change MST or not?

d$.(5$ pts$)$ Adding the same positive value to every edge of $G$ can change shortest path between two vertices or not?

For all question above, do not leave your answer as YES or NO$.$ Discuss the situation with examples $($by drawing simple graphs$).$ If you can, prove your answer by contradiction or counter$-$examples. For some questions, answer may change at different conditions. If so$,$ please discuss the answer separately for all different cases.