The $p$-spectral radius of the Laplacian

The $p$-spectral radius of a graph $G=(V,E)$ with adjacency matrix $A$ is defined as ${\lambda }^{(p)}(G)=\max \{x^{T}Ax:\mathrm{\parallel }x{\mathrm{\parallel }}_p=1\}$. This parameter shows remarkable connections with graph invariants, and has been used to generalize some extremal problems. In this work, we extend this approach to the Laplacian matrix $L$, and define the $p$-spectral radius of the Laplacian as ${\mu }^{(p)}(G)=\max \{x^{T}Lx:\mathrm{\parallel }x{\mathrm{\parallel }}_p=1\}$. We show that ${\mu }^{(p)}(G)$ relates to invariants such as maximum degree and size of a maximum cut. We also show properties of ${\mu }^{(p)}(G)$ as a function of $p$, and a upper bound on ${\max }_{G\text{\hspace{0.17em}⁣}:\mathrm{\mid }V(G)\mathrm{\mid }=n}{\mu }^{(p)}(G)$ in terms of $n=\mathrm{\mid }V\mathrm{\mid }$ for $p\ge 2$, which is attained if $n$ is even.