发表机构
Kyungpook National University（庆北国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了图辫群同调中所有有限阶挠元的结构，给出了显式代表元、最小图条件及无挠性判定，并证明了奇挠元首次出现的具体位置。
AI 中文摘要
我们确定了$H_{m-1}(\mathbb{B}_mK_{m+1,m+r-1};\mathbb{Z})$在$m\ge2$和$r\ge0$时的挠子群：任意系数的最高同调是无符号子集包含矩阵的核，其积分对角形式决定了所有初等和。每个有限阶都会出现，并给出显式代表元。广义theta类张成一个包含矩阵余核的嵌入副本，其中包含所有挠元；当$r\ge m$时，它们生成挠元，每个阶为$\operatorname{lcm}(1,\ldots,m)$。对于每个素幂$q$和$m\ge q$，图$K_{m+1,m+q-1}$在子图序中对于$H_{m-1}(\mathbb{B}_m)$中的$q$阶挠元是最小的。特别地，奇挠元首次出现在$H_2(\mathbb{B}_3K_{4,5})\cong\mathbb{Z}^{155}\oplus(\mathbb{Z}/2)^4\oplus\mathbb{Z}/3$中，且$K_{4,5}$的没有真子图在$H_2(\mathbb{B}_3)$中有奇挠元。对于任意部分大小，我们给出了$H_m(\mathbb{B}_mK_{a,b};\mathbb{Q})$在顶点置换下的无重分解，并证明了当$a,b\ge2m-1$且$p\ge m$为奇素数时，$H_{m-1}(\mathbb{B}_mK_{a,b};\mathbb{Z})$没有$p$-初等挠元。显式的$q$阶类在图的第二部分每次扩大时保持其阶，而当$m=q=p$为奇素数时，它被第一部分的特定扩大所消灭。
英文摘要
We determine the torsion subgroup of $H_{m-1}(\mathbb{B}_mK_{m+1,m+r-1};\mathbb{Z})$ for $m\ge2$ and $r\ge0$: top homology with arbitrary coefficients is the kernel of an unsigned subset-inclusion matrix, and its integral diagonal form determines all primary summands. Every finite order occurs, with explicit representatives. Generalized theta classes span an embedded copy of the cokernel of the inclusion matrix, containing all torsion; for $r\ge m$ they generate the torsion, each of order $\operatorname{lcm}(1,\ldots,m)$. For every prime power $q$ and $m\ge q$, the graph $K_{m+1,m+q-1}$ is minimal in the minor order for order-$q$ torsion in $H_{m-1}(\mathbb{B}_m)$. In particular, odd torsion first appears in $H_2(\mathbb{B}_3K_{4,5})\cong\mathbb{Z}^{155}\oplus(\mathbb{Z}/2)^4\oplus\mathbb{Z}/3$, and no proper minor of $K_{4,5}$ has odd torsion in $H_2(\mathbb{B}_3)$. For arbitrary part sizes, we give a multiplicity-free decomposition of $H_m(\mathbb{B}_mK_{a,b};\mathbb{Q})$ under vertex permutations and prove that $H_{m-1}(\mathbb{B}_mK_{a,b};\mathbb{Z})$ has no $p$-primary torsion when $a,b\ge2m-1$ and $p\ge m$ is an odd prime. The explicit order-$q$ class retains its order under every enlargement of the second part of the graph, while for $m=q=p$ an odd prime it is killed by a specified enlargement of the first part.
