For a simple graph $G$, let $n$ and $m$ denote the number of vertices and edges in $G$, respectively. The Erd\H{o}s-Gallai theorem for paths states that in a simple $P_k$-free graph, $m \leq \frac{n(k-1)}{2}$, where $P_k$ denotes a path with length $k$ (that is, with $k$ edges). In this paper, we generalize this result as follows: For each $v \in V(G)$, let $p(v)$ be the length of the longest path that contains $v$. We show that \[m \leq \sum_{v \in V(G)} \frac{p(v)}{2}\] The Erd\H{o}s-Gallai theorem for cycles states that in a simple graph $G$ with circumference (that is, the length of the longest cycle) at most $k$, we have $m \leq \frac{k(n-1)}{2}$. We strengthen this result as follows: For each $v \in V(G)$, let $c(v)$ be the length of the longest cycle that contains $v$, or $2$ if $v$ is not part of any cycle. We prove that \[m \leq \left( \sum_{v \in V(G)} \frac{c(v)}{2} \right) - \frac{c(u)}{2}\] where $c(u)$ denotes the circumference of $G$. \newline Furthermore, we characterize the class of extremal graphs that attain equality in these bounds.

翻译：对于简单图$G$，记$n$与$m$分别为其顶点数与边数。关于路径的Erdős-Gallai定理指出：在不含$P_k$的简单图中，有$m \leq \frac{n(k-1)}{2}$，其中$P_k$表示长度为$k$（即包含$k$条边）的路径。本文将该结果推广如下：对每个$v \in V(G)$，令$p(v)$为经过$v$的最长路径的长度。我们证明\[m \leq \sum_{v \in V(G)} \frac{p(v)}{2}\]。关于圈的Erdős-Gallai定理指出：在周长（即最长圈的长度）不超过$k$的简单图$G$中，有$m \leq \frac{k(n-1)}{2}$。我们强化该结果如下：对每个$v \in V(G)$，令$c(v)$为经过$v$的最长圈的长度；若$v$不属于任何圈，则令$c(v)=2$。我们证明\[m \leq \left( \sum_{v \in V(G)} \frac{c(v)}{2} \right) - \frac{c(u)}{2}\]，其中$c(u)$表示图$G$的周长。\newline 此外，我们刻画了达到这些界等号的极图类。