meta - Spanning Tree Graph -


i have question spanning tree , not entirely sure how tackle properly.

a connected graph g= (v, e) withn=|v|vertices can have many different spanningtrees, each exactlyn−1 edges. relationships among these spanning trees can berepresented meta-graph sg. each vertex ofsgcorresponds 1 of spanning treesofg . there edge between 2 vertices of sg if corresponding spanning trees ti andtj have n−2 common edges ing(so ti has 1 edge ofg not intj , tj has 1 edge of g not in ti).

(a) prove meta-graph sg of connected graphg connected.

(b) suppose edges of g have positive integer weights. weights not necessar-ily distinct, there can more 1 minimum spanning tree (mst). vertices of sg correspond msts of g , other vertices correspond spanning trees of g not have minimum weight. prove subgraph of sg inducedby vertices of sg corresponding msts of g connected.

answer: not entirely sure how solve problem , correct argument proove a) , b)

this have far part a) suppose metagraph of g not connected means there @ least 2 spanning trees @ least 2 different edges not possible since spanning trees cover possible tree formation of connected graph.so there must spanning tree has 1 different edge other spanning tree.

could please me question?


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