The Number of Spanning Trees in Generalized Complete Multipartite Graphs of Fan-Type

Let Kk,n be a complete k-partite graph of order n and let K k,n be a generalized complete k-partite graph of order n spanned by the fan set F = {Fn1 , Fn2 , · · · , Fnk}, where n = {n1, n2, · · · , nk} and n = n1 + n2 + · · ·+ nk for 1 6 k 6 n. In this paper, we get the number of spanning trees in Kk,n to be t(Kk,n) = n k−2 k ∏ i=1 (n− ni)i. and the number of spanning trees in K k,n to be t(K k,n) = n 2k−2 k ∏ i=1 αi i − β ni−1 i αi − βi where αi = (di + √ di − 4)/2 and βi = (di − √ di − 4)/2,di = n − ni + 3. In particular, K1,n = K c n with t(K1,n) = 0, Kn,n = Kn with t(Kn,n) = n n−2 which is just the Cayley’s formula and K 1,n = Fn with t(K F 1,n) = (α n−1 − β)/ √ 5 where α = (3 + √ 5)/2 and β = (3− √ 5)/2 which is just the formula given by Z.R.Bogdanowicz in 2008.