Signed spectral Turań type theorems

A signed graph Σ=(G,σ) is a where the function σ assigns either 1 or −1 to each edge of simple G. The adjacency matrix Σ, denoted by A(Σ), defined canonically. In recent paper, Wang et al. extended spectral bounds Hoffman and Cvetković for chromatic number graphs. They proposed an open problem related balanced clique largest eigenvalue graph. We solve strengthened version this problem. As byproduct, we give alternate proofs some known classical least eigenvalues unsigned extend Turán's inequality Besides, study Bollobás Nikiforov conjecture graphs show that need not be true Nevertheless, holds under assumptions. Finally, relationships between walks