On the number of spanning trees of Knm±G graphs

TheKn-complement of a graph G, denoted byKn − G, is defined as the graph obtained from the complete graph Kn by removing a set of edges that span G; if G hasn vertices, thenKn − G coincides with the complement G of the graphG. In this paper we extend the previous notion and derive deter minant based formulas for the number of spanning trees of graphs of the form K n ± G, whereK m n is the complete multigraph on vertices with exactlym edges joining every pair of vertices and G is a multigraph spanned by a set of edges of K n ; the graphK m n +G (resp. K n − G) is obtained fromK m n by adding (resp. removing) the edges of G. Moreover, we derive determinant based formulas for graphs that result fromK n by adding and removing edges of multigraphs spanned by sets o f dges of the graphK n . We also prove closed formulas for the number of spanning tre e of graphs of the formK m n ±G, where G is (i) a complete multipartite graph, and (ii) a multi-star g raph. Our results generalize previous results and extend the family of graphs admitting formulas for the number of the ir spanning trees.