(Solution) - a Consider the subgraph of G in Fig 12 54 induced -(2025 Original AI-Free Solution)

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(a) Consider the subgraph of G (in Fig. 12.54) induced by the vertices a, b, c, d. This graph is called a kite. How many nonidentical (though some may be isomorphic) spanning trees are there for this kite?
(b) How many nonidentical (though some may be isomorphic) spanning trees of G do not contain the edge {c, h}?
(c) How many nonidentical (though some may be isomorphic) spanning trees of G contain all four of the edges {c, h}, {g, k], {l, p], and {d, o}?
(d) How many nonidentical (though some may be isomorphic) spanning trees exist for G?
(e) We generalize the graph G as follows. For n ? 2, start with a cycle on the In vertices v1, v2, ... , v2n-1, v2n. Replace each of the n edges {v1, v2], {v3, v4},... , {v2n-1, v2n} with a (labeled) kite so that the resulting graph is 3-regular. (The case for n = 4 appears in Fig. 12.54.) How many nonidentical (though some may be isomorphic) spanning trees are there for this graph?