Let mn Z mn 1 the complete bipartite graph Kmn is the graph

Let m,n Z, m,n 1. the complete bipartite graph Km,n is the graph with set of vertices V = V1V2, such that |V1| = m and |V2| = n, and with set of
edges {{v1,v2}: v1V1 , v2 V2}. For which values of (m, n) is Km,n a tree?

Solution

tree cannot have a cycle

Since the graph is complete bipartite,

every vertex of V1 has an edge with every vertex of V2

if both of V1 and V2 have atleast two edges

then there is a cycle

u1, u2, v1, v2, u1

where v1,u1 is in V1

and u2,v2 in V2

So either of m or n =1

that is graph is a star graph