Localization of spectral Turán theorems for signed graphs

Abstract

In this paper, we extend localized Turán theorems to signed graphs and study the corresponding spectral Turán problems. Firstly, we establish a vertex-localized Turán-type inequality for signed graphs and characterize the extremal graphs reaching this bound. Secondly, we derive a sharp upper bound for the largest eigenvalue of a signed graph via localized Turán parameters, and identify the extremal graphs attaining equality. Finally, we generalize a walk-based local spectral Turán inequality to signed graphs. Our results generalize and improve several existing theorems for unsigned and signed graphs.