Question

4.32 Show that there is only one positive integer k such that no graph contains exactly k spanning trees.

Answer 1

Papo No. As we know that, by graph we mean a connected undisec ted graph. So, for proof or to show our result, look at the k-th regulas I bolygon , K23. It has k nodes and k edges, its spanning bree is all the edges minus 1: It has exactly K spanning trees. lie, simply we can say ff G is a finite connected graph with n vertices then all spanning tees of G must have not edges. Also, adding some edges and hodes to a graph would not reduce its number of spanning trees. A groph has exactly one spanning

tsee iff it is a tree (un - cyclic), otherwise it has a cycle of length K23 and by the preceding, it has at least k spanning trees. For k=2, let us show that a graph cannot have exactly two distinct spanning trees. We know that, union of two spanning trees contains a cycle le contains too many edges to be a free , Cycles have length greater than 2. Removing any edge from a cycle leaves a connected graph, so the union of the two spanning trees has at least 3 spanning trees, each of which is a spanning tree of the original graph. Hence this shows the required result. Note: In example, two spanning trees. for a graph in are shown as Teacher's Son.