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

Extremal problems concerning the number of complete subgraphs have a long story in extremal graph theory. Let $k_s(G)$ be the number of $s$-cliques in a graph $G$ and $m=\left(\genfrac{}{}{0px}{}{r_m}{s}\right)+t_m$, where $0\le t_m\le r_m$. Edr\H{o}s showed that $k_s(G)\le \left(\genfrac{}{}{0px}{}{r_m}{s}\right)+\left(\genfrac{}{}{0px}{}{t_m}{s-1}\right)$ over all graphs of size $m$ and order $n\ge r_m+1$. %Clearly, $K_{r_m}^{t_m}\cup (n-r_m-1)K_1$ is an extremal graph, where $K_{r_m}^{t_m}$ is the graph by joining a new vertex to $t_m$ vertices of $K_{r_m}$. It is natural to consider an improvement in connected situation: what is the maximum number of $s$-cliques over all connected graphs of size $m$ and order $n$? In this paper, the sharp upper bound 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 an application, this result can be used to solve a question on spectral moment.