On the Spectral Radius of Unicyclic Graphs with \(n\) Vertices and Edge Independence Number \(q\)

Abstract

We study the spectral radius of unicyclic graphs with \(n\) vertices and edge independence number \(q\). In this paper, we show that of all unicyclic graphs with \(n\) vertices and edge independence number \(q\), the maximal spectral radius is obtained uniquely at \(\Delta_n(q)\), where \(\Delta_n(q)\) is a graph on \(n\) vertices obtained from the cycle \(C_3\) by attaching \(n – 2q + 1\) pendant edges and \(q – 2\) paths of length \(2\) at one vertex.