Let $G$ be a connected graph and $T$ a spanning tree of $G$. Let $\rho(G)$ denote the adjacency spectral radius of $G$. The $k$-excess of a vertex $v$ in $T$ is defined as $\max\{ 0,d_T(v)-k\}$. The total $k$-excess $\mbox{ te}(T,k)$ is defined by $\mbox{ te}(T,k)=\sum\limits_{ v\in V(T)}{ \max\{ 0,d_T(v)-k\}}$. A tree $T$ is said to be a $k$-tree if $d_T(v)\leq k$ for any $v\in V(T)$, that is to say, the maximum degree of a $k$-tree is at most $k$. In fact, $T$ is a spanning $k$-tree if and only if $\mbox{ te}(T,k)=0$. This paper studies a generalization of spanning $k$-trees using a concept called total $k$-excess and proposes a lower bound for $\rho(G)$ in a connected graph $G$ to ensure that $G$ contains a spanning tree $T$ with $\mbox{ te}(T,k)\leq b$, where $k$ and $b$ are two nonnegative integers with $k\geq\max\{ 5,b+3\}$ and $(b,k)\neq(2,5)$.
Categories: math.CO