Signed spectral Turań type theorems

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