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 ${\mathrm{\Delta }}_n(q)$, where ${\mathrm{\Delta }}_n(q)$ 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.