A Tur\'an-type problem on degree sequence

Given $p\geq 0$ and a graph $G$ whose degree sequence is $d_1,d_2,\ldots,d_n$, let $e_p(G)=\sum_{i=1}^n d_i^p$. Caro and Yuster introduced a Tur\'an-type problem for $e_p(G)$: given $p\geq 0$, how large can $e_p(G)$ be if $G$ has no subgraph of a particular type. Denote by $ex_p(n,H)$ the maximum value of $e_p(G)$ taken over all graphs with $n$ vertices that do not contain $H$ as a subgraph. Clearly, $ex_1(n,H)=2ex(n,H)$, where $ex(n,H)$ denotes the classical Tur\'an number, i.e., the maximum number of edges among all $H$-free graphs with $n$ vertices. Pikhurko and Taraz generalize this Tur\'an-type problem: let $f$ be a non-negative increasing real function and $e_f(G)=\sum_{i=1}^n f(d_i)$, and then define $ex_f(n,H)$ as the maximum value of $e_f(G)$ taken over all graphs with $n$ vertices that do not contain $H$ as a subgraph. Observe that $ex_f(n,H)=ex(n,H)$ if $f(x)=x/2$, $ex_f(n,H)=ex_p(n,H)$ if $f(x)=x^p$. Bollob\'as and Nikiforov mentioned that it is important to study concrete functions. They gave an example $f(x)=\phi(k)={x\choose k}$, since $\sum_{i=1}^n{d_i\choose k}$ counts the $(k+1)$-vertex subgraphs of $G$ with a dominating vertex. Denote by $T_r(n)$ the $r$-partite Tur\'an graph of order $n$. In this paper, using the Bollob\'as--Nikiforov's methods, we give some results on $ex_{\phi}(n,K_{r+1})$ $(r\geq 2)$ as follows: for $k=1,2$, $ex_\phi(n,K_{r+1})=e_\phi(T_r(n))$; for each $k$, there exists a constant $c=c(k)$ such that for every $r\geq c(k)$ and sufficiently large $n$, $ex_\phi(n,K_{r+1})=e_\phi(T_r(n))$; for a fixed $(r+1)$-chromatic graph $H$ and every $k$, when $n$ is sufficiently large, we have $ex_\phi(n,H)=e_\phi(n,K_{r+1})+o(n^{k+1})$.