Maxima of the Q-index: degenerate graphs

Let $G$ be a $k$-degenerate graph of order $n\mathrm{.}$ It is well-known that G\has no more edges than $S_{n,k},$ the join of a complete graph of order $k$ and an independent set of order $n-k\mathrm{.}$ In this note it is shown that $S_{n,k}$ is extremal for some spectral parameters of $G$ as well. More precisely, letting $\mu \left(H\right)$ and $q\left(H\right)$ denote the largest eigenvalues of the adjacency matrix and the signless Laplacian of a graph $H,$ the inequalities $$\mu \left(G\right)<\mu \left(S_{n,k}\right)\text{ and }q\left(G\right)<q\left(S_{n,k}\right)$$ hold, unless $G=S_{n,k}$. The latter inequality is deduced from the following general bound, which improves some previous bounds on $q\left(G\right)$: If $G$ is a graph of order $n$, with $m$ edges, with maximum degree $\mathrm{\Delta }$ and minimum degree $\delta ,$ then $$q\left(G\right)\le \min \{2\mathrm{\Delta },\frac{1}{2}(\mathrm{\Delta }+2\delta -1+\sqrt{{(\mathrm{\Delta }+2\delta -1)}^2+16m-8(n-1+\mathrm{\Delta })\delta })\}\mathrm{.}$$ Equality holds if and only if $G$ is regular or $G$ has a component of order $\mathrm{\Delta }+1$ in which every vertex is of degree $\delta$ or $\mathrm{\Delta },$ and all other components are $\delta$-regular.