On the Spectral Radius of Quasi-Unicycle Graphs

Abstract

A connected graph \(G = (V, E)\) is called a quasi-unicycle graph if there exists \(v_0 \in V\) such that \(G – v_0\) is a unicycle graph. Denote by \(\mathcal{G}(n, d_0)\) the set of quasi-unicycle graphs of order \(n\) with the vertex \(v_0\) of degree \(d_0\) such that \(G – v_0\) is a unicycle graph. In this paper, we determine the maximum spectral radii of quasi-unicycle graphs in \(\mathcal{G}(n, d_0)\).