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Extremal problems concerning the number of complete subgraphs have a longstory in extremal graph theory. Let $k_s(G)$ be the number of $s$-cliques in agraph $G$ and $m={{r_m}\choose s}+t_m$, where $0\le t_m\leq r_m$. Edr\H{o}sshowed that $k_s(G)\le {{r_m}\choose s}+{{t_m}\choose{s-1}}$ over all graphs ofsize $m$ and order $n\geq r_m+1$. %Clearly, $K_{r_m}^{t_m}\cup (n-r_m-1)K_1$ isan extremal graph, where $K_{r_m}^{t_m}$ is the graph by joining a new vertexto $t_m$ vertices of $K_{r_m}$. It is natural to consider an improvement inconnected situation: what is the maximum number of $s$-cliques over allconnected graphs of size $m$ and order $n$? In this paper, the sharp upperbound of $k_s(G)$ is obtained and extremal graphs are completely characterized.The technique and the bound are different from those in general case. As anapplication, this result can be used to solve a question on spectral moment.

The maximum number of s-cliques in connected graphs and its application to spectral moment