On the Spectral Radius of Unicyclic Graphs with Fixed Maximum Degree

Abstract

Let $\mathrm{\Delta }(G)$ be the maximum degree of a graph $G$, and let $\mathcal{U}(n,\mathrm{\Delta })$ be the set of all unicyclic graphs on $n$ vertices with fixed maximum degree $\mathrm{\Delta }$. Among all the graphs in $\mathcal{U}(n,\mathrm{\Delta })$ ($\mathrm{\Delta }\ge \frac{n+3}{2}$), we characterize the graph with the maximal spectral radius. We also prove that the spectral radius of a unicyclic graph $G$ on $n$ ($n\ge 30$) vertices strictly increases with its maximum degree when $\mathrm{\Delta }(G)\ge \lceil \frac{7n}{9}\rceil +1$.