Signed Complete Graphs with Exactly m Non-negative Eigenvalues

Let ( G , σ ) be a signed graph, where G is the underlying simple graph and σ : E ( G ) ⟶ { - , + } is the sign function on the edges of G . Let ( K n , H - ) be a signed complete graph whose negative edges induce a subgraph H . In this paper, we show that if ( K n , H - ) has exactly m non-negative eigenvalues (including their multiplicities), then H has at most m - 1 connected components. In particular, if H is a non-spanning subgraph of K n with m - 1 connected components, then each connected component of H is a complete bipartite graph. Next, we characterize all signed graphs ( K n , σ ) with exactly one or two non-negative eigenvalues. Let H be a non-spanning subgraph of K n . Then, we characterize ( K n , H - ) with exactly three non-negative eigenvalues, where H is not connected or H is a tree. Also, it is shown that if H is a connected non-spanning subgraph of K n and H has exactly two negative eigenvalues, then ( K n , H - ) has exactly three non-negative eigenvalues.