Vertex-Based Localization of Tur\'{a}n's Theorem

Let $G$ be a simple graph with $n$ vertices and $m$ edges. According to Tur\'{a}n's theorem, if $G$ is $K_{r+1}$-free, then $m \leq |E(T(n, r))|,$ where $T(n, r)$ denotes the Tur\'{a}n graph on $n$ vertices with a maximum clique of order $r$. A limitation of this statement is that it does not give an expression in terms of $n$ and $r$. A widely used version of Tur\'{a}n's theorem states that for an $n$-vertex $K_{r+1}$-free graph, $m \leq \left\lfloor \frac{n^2(r-1)}{2r} \right\rfloor.$ Though this bound is often more convenient, it is not the same as the original statement. In particular, the class of extremal graphs for this bound, say $\mathcal{S}$, is a proper subset of the set of Tur\'{a}n graphs. In this paper, we generalize this result as follows: For each $v \in V(G)$, let $c(v)$ be the order of the largest clique that contains $v$. We show that $$m \leq \left\lfloor\frac{n}{2}\sum_{v\in V(G)}\frac{c(v)-1}{c(v)}\right\rfloor$$ Furthermore, we characterize the class of extremal graphs that attain equality in this bound. Interestingly, this class contains two extra non-Tur\'{a}n graphs other than the graphs in $\mathcal{S}$.