Index perturbation of signed graphs

Abstract

Let $\hat{I}“=(G,\ddot{I}ƒ)$ be a signed graph and $v$ a non-isolated vertex of $\hat{I}“$. Let $\hat{I}“-v$ denote the graph obtained by deleting the vertex $v$ together with all signed edges incident to it from $\hat{I}“$, and $d_{\hat{I}}“(v)$ the degree of $v$ in $\hat{I}“$. In this paper, we prove that the largest eigenvalue $\hat{I}{»}_1(\hat{I}“)$ of $\hat{I}“$ satisfies \[ λ_1(Γ) \le \sqrt{λ_1^2(Γ- v) + 2d_Γ(v) - 1}, \] and we also present a refined version of this bound. Moreover, we characterize the extremal signed graphs achieving equality when $\hat{I}“$ is connected and $d_{\hat{I}}“(v)\ge 2$, which are switching equivalent to the balanced complete signed graph.