The number of spanning trees in $$K_n$$ -complement of a bipartite graph

Abstract

For a subgraph G of a complete graph \(K_n\), the \(K_n\)-complement of G, denoted by \(K_n-G\), is the graph obtained from \(K_n-G\) by removing all the edges of G. In this paper, we express the number of spanning trees of the \(K_n\)-complement \(K_n-G\) of a bipartite graph G in terms of the determinant of the biadjcency matrices of all induced balanced bipartite subgraphs of G, which are nonsingular, and we derive formulas of the number of spanning trees of \(K_n-G\) for various important classes of bipartite graphs G, some of which generalize some previous results.