On signed graphs whose spectral radius does not exceed $\sqrt{2+\sqrt{5}}$

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Abstract

The Hoffman program with respect to any real or complex square matrix $M$ associated to a graph $G$ stems from Hoffman's pioneering work on the limit points for the spectral radius of adjacency matrices of graphs does not exceed $\sqrt{2+\sqrt{5}}$. A signed graph $\dot{G}=(G,\sigma)$ is a pair $(G,\sigma),$ where $G=(V,E)$ is a simple graph and $\sigma: E(G)\rightarrow \{+1,-1\}$ is the sign function. In this paper, we study the Hoffman program of signed graphs. Here, all signed graphs whose spectral radius does not exceed $\sqrt{2+\sqrt{5}}$ will be identified.