发表机构
College of Finance and Mathematics, West Anhui University（安徽西皖大学金融与数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究分析树的广义同谱伴侣的连通分支约束，推导双星图 $D(2m,m+1)$ 为广义谱确定而非邻接谱确定的条件，并给出显式无限家族的 DGS 非 DS 图。
AI 中文摘要
我们研究与主特征值相关的不可约因子如何约束广义同谱伴侣的连通分支。设 $M_G(x):=\boldsymbol{\rm 主多项式}$，其中乘积遍历不同的主特征值；记 $\boldsymbol{\rm 特征多项式}$，其中 $f_i$ 是 $\boldsymbol{\rm 有理数域}$ 上不同的首一不可约多项式。定义 $\boldsymbol{\rm 主因子索引集}$，并令 $\boldsymbol{\rm 主因子重数和}$。对于图 $X$，记 $c(X)$ 和 $\beta(X)$ 分别为其连通分支数和圈秩。我们证明，与图 $G$ 具有相同特征多项式和主多项式的任意图 $H$ 满足 $c(H)\boldsymbol{\rm \text{≤} \boldsymbol{\rm κ_m}(G)}$。由此可得，若 $T$ 是树且 $H$ 与 $T$ 广义同谱，则 $\beta(H)=c(H)-1 \boldsymbol{\rm \text{≤} \boldsymbol{\rm κ_m}(T)-1}$。我们进一步研究 $\boldsymbol{\rm κ_m}(T)=2$ 的情形：不连通的广义同谱伴侣必为一棵树与一个连通二分单圈图的并，其两个分支冠由 $T$ 关于两个不可约主因子的冠的典范部分分式分解唯一确定；因此，这些因子冠必须可实现为实际图分支的冠，其 Laurent 系数必须为满足低阶游走恒等式的非负整数。这些可实现性条件，加上对单圈分支的匹配限制，产生了有效的树强制障碍。作为应用，我们证明当 $m\boldsymbol{\rm \text{≥} 2}$ 且 $m$ 和 $2m+2$ 均非完全平方数时，双星图 $D(2m,m+1)$ 由其广义谱确定，但不由其邻接谱确定；特别地，双星图 $D(8t+4,4t+3) (t\boldsymbol{\rm \text{≥} 0})$ 是显式的无限家族图，属于广义谱确定（DGS）但非邻接谱确定（DS）。
英文摘要
We study how irreducible factors associated with main eigenvalues constrain the connected components of generalized cospectral mates. Let $M_G(x)$ be the main polynomial of a graph $G$, and let $κ_{\mathrm m}(G)$ denote the sum of the multiplicities in $ϕ_G(x)$ of the irreducible factors dividing $M_G(x)$. We prove that every graph $H$ with the same characteristic polynomial and main polynomial as $G$ satisfies $c(H)\leq κ_{\mathrm m}(G)$. Consequently, if $T$ is a tree and $H$ is generalized cospectral with $T$, then $β(H)=c(H)-1\leq κ_{\mathrm m}(T)-1$. We develop the case $κ_{\mathrm m}(T)=2$ in detail. Any disconnected generalized cospectral mate is the union of a tree and a connected bipartite unicyclic graph, and the coronals of its two components are uniquely prescribed by the canonical decomposition of the coronal of $T$ with respect to the two irreducible main factors. This turns the existence of a disconnected mate into a component-realizability problem. Factor moments yield realizability and tree-forcing obstructions, while matching data provide complementary characteristic-polynomial information. In particular, the unique cycle of any disconnected mate has length at least $6$, and finitely many matching identities, together with component-coronal realizability, certify generalized cospectrality under an explicit degree bound. As an application, we show that the double star $D(2m,m+1)$ is DGS but not DS whenever $m\geq2$ and neither $m$ nor $2m+2$ is a perfect square. In particular, $D(8t+4,4t+3)$, $t\geq0$, gives an explicit infinite family of such graphs.
Comments20 pages, 1 figure. Substantially revised, especially the treatment of matching restrictions and finite certification in the two-factor case
