Signed spectral Turan type theorems

A signed graph sigma = (G, sigma) is a graph where the function sigma assigns either 1 or -1 to each edge of the simple graph G. The adjacency matrix of sigma, denoted by A(sigma), is defined canonically. In a recent paper, Wang et al. extended the spectral bounds of Hoffman and Cvetkovic for the chromatic number of 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 unsigned graphs. We extend the Turan's inequality for the signed graphs. Besides, we study the Bollobas 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.(c) 2023 Elsevier Inc. All rights reserved.