On the difference between the spectral radius and the maximum degree of graphs

Let $G$ be a graph with the eigenvalues $\lambda_1(G)\geq\cdots\geq\lambda_n(G)$. The largest eigenvalue of $G$, $\lambda_1(G)$, is called the spectral radius of $G$. Let $\beta(G)=\Delta(G)-\lambda_1(G)$, where $\Delta(G)$ is the maximum degree of vertices of $G$. It is known that if $G$ is a connected graph, then $\beta(G)\geq0$ and the equality holds if and only if $G$ is regular. In this paper we study the maximum value and the minimum value of $\beta(G)$ among all non-regular connected graphs. In particular we show that for every tree $T$ with $n\geq3$ vertices, $n-1-\sqrt{n-1}\geq\beta(T)\geq 4\sin^2(\frac{\pi}{2n+2})$. Moreover, we prove that in the right side the equality holds if and only if $T\cong P_n$ and in the other side the equality holds if and only if $T\cong S_n$, where $P_n$ and $S_n$ are the path and the star on $n$ vertices, respectively.