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

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 $\sigma :E(G)\to \{+1,-1\}$ 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.

arXiv comment33 pages, 20 figures