Extremal problems for the p-spectral radius of graphs

The $p$-spectral radius of a graph $G\$of order $n$ is defined for any real number $p\geq1$ as $$\lambda^{\left( p\right) }\left( G\right) =\max\left\{ 2\sum_{\{i,j\}\in E\left( G\right) \ }x_{i}x_{j}:x_{1},\ldots,x_{n}\in\mathbb{R}\text{ and }\left\vert x_{1}\right\vert ^{p}+\cdots+\left\vert x_{n}\right\vert ^{p}=1\right\} .$$ The most remarkable feature of $\lambda^{\left( p\right) }$ is that it seamlessly joins several other graph parameters, e.g., $\lambda^{\left( 1\right) }$ is the Lagrangian, $\lambda^{\left( 2\right) }$ is the spectral radius and $\lambda^{\left( \infty\right) }/2$ is the number of edges. This paper presents solutions to some extremal problems about $\lambda^{\left( p\right) }$, which are common generalizations of corresponding edge and spectral extremal problems. Let $T_{r}\left( n\right)$ be the $r$-partite Tur\'{a}n graph of order $n.$ Two of the main results in the paper are: (I) Let $r\geq2$ and $p>1.$ If $G$ is a $K_{r+1}$-free graph of order $n,$ then $$\lambda^{\left( p\right) }\left( G\right) <\lambda^{\left( p\right) }\left( T_{r}\left( n\right) \right) ,$$ unless $G=T_{r}\left( n\right) .$ (II) Let $r\geq2$ and $p>1.$ If $G\$is a graph of order $n,$ with $$\lambda^{\left( p\right) }\left( G\right) >\lambda^{\left( p\right) }\left( T_{r}\left( n\right) \right) ,$$ then $G$ has an edge contained in at least $cn^{r-1}$ cliques of order $r+1,$ where $c$ is a positive number depending only on $p$ and $r.$Comment: 21 pages. Some minor corrections in v