Counting matchable spanning trees of join graphs

A spanning tree T of a graph G is called a matchable spanning tree if T has a perfect matching. Let t′(G) denote the number of matchable spanning trees in G. In 1984, Simion first investigated the enumerative problem concerning matchable spanning trees in the complete graph K2n and proved that t′(K2n)=(2n)n−2(2n)!/n!. Let G∨G′ be the join of two vertex-disjoint graphs G and G′. In this paper, we show that t′(G∨Knc)=(n−1)!∏i=1n−1(2n+μi(G)) for any graph G of order n, where {μ1(G)≥μ2(G)≥⋯≥μn−1(G)≥μn(G)=0} is the Laplacian spectrum of G, and Knc is the complement of the complete graph Kn. Moreover, we prove that t′(Kr∨Ksc)=2×r!(r+s)(r−s−2)/2(2r+s)s−1/[(r−s)/2]! for r≥s and r≡s(mod2).