On the number of spanning trees of Knm±G graphs

The Kn-complement of a graph G, denoted by Kn − G, is defined as the graph obtained from the complete graph Kn by removing a set of edges that span G; if G has n vertices, then Kn − G coincides with the complement G of the graph G. In this paper we extend the previous notion and derive determinant based formulas for the number of spanning trees of graphs of the form K n ± G, where K m n is the complete multigraph on n vertices with exactly m edges joining every pair of vertices and G is a multigraph spanned by a set of edges of K n ; the graph K m n +G (resp. K n −G) is obtained from K m n by adding (resp. removing) the edges of G. Moreover, we derive determinant based formulas for graphs that result from K n by adding and removing edges of multigraphs spanned by sets of edges of the graph K n . We also prove closed formulas for the number of spanning tree of graphs of the form K m n ±G, where G is (i) a complete multipartite graph, and (ii) a multi-star graph. Our results generalize previous results and extend the family of graphs admitting formulas for the number of their spanning trees.