发表机构
Faculty of Mathematics and Statistics, Central China Normal University（华中师范大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究确定了2-连通图中最大化各大小独立集数量的唯一极值图，即团阶差至多1的平衡环链，并将连通图的极值结果推广至2-连通情形。
AI 中文摘要
对于每个 $\alpha\ge3$ 和所有足够大的 $n$，我们确定了一个唯一的图，它在所有具有独立数 $\alpha$ 的 $n$ 顶点 2-连通图中，同时最大化每个大小的独立集的数量。该图在同构意义下唯一，并且是 $\alpha$ 个团（其阶数至多相差 1）的不交并的平衡环链。更精确地，对于每个 $3\le\beta\le\alpha$，最大化独立 $\beta$-集数量的图恰好是那些团大小循环词为 $\lfloor\beta/2\rfloor$-平衡的环链。这些结果将连通图的相应极值结果推广到 2-连通情形。证明结合了广义 Turán 型团计数和 2-连通图的边极值理论，以及系数式平衡交换论证。该交换还表明，长度为 $r$ 的最短不平衡首先在度数 $2r$ 处影响独立集计数。
英文摘要
For every $α\ge3$ and all sufficiently large $n$, we identify a single graph that simultaneously maximizes the number of independent sets of every size among all $n$-vertex $2$-connected graphs with independence number $α$. This graph is unique up to isomorphism and is a balanced cycle-linking of the disjoint union of $α$ cliques whose orders differ by at most one. More precisely, for each $3\leβ\leα$, the graphs maximizing the number of independent $β$-sets are exactly the cycle-linkings whose clique-size cyclic words are $\lfloorβ/2\rfloor$-balanced. These results extend the corresponding extremal result for connected graphs to the $2$-connected setting. The proof combines generalized Turán-type clique counting and the edge-extremal theory of $2$-connected graphs with a coefficientwise balancing-switch argument. The switch also shows that a shortest imbalance of length $r$ first affects the independent-set count in degree $2r$.
