凝聚集的补集
Complements of coalescing sets
摘要：
我们考虑$qD+A$形式的矩阵，其中$D$是度的对角矩阵，$A$是邻接矩阵，并且$q$是固定值。给定一个图$H$和$B\subseteq V(G)$，我们称之为一个合并对$(H,B)$，我们导出了一个特征多项式的公式，其中同一根图$G$的一个副本通过根连接到$B$的\emph{each}顶点.进一步证明了如果$(H_1,B_1)$和$(H_2,B_2)$是两个对任何可能的根图$G$共谱的结合对，那么对任何可能的根图$G$，$(H_1,V(H_1)\setminus B_1)$和$(H_2,V(H_2)\setminus B_2)$也总是共谱的.
英文摘要：
We consider matrices of the form $qD+A$, with $D$ being the diagonal matrix of degrees, $A$ being the adjacency matrix, and $q$ a fixed value. Given a graph $H$ and $B\subseteq V(G)$, which we call a coalescent pair $(H,B)$, we derive a formula for the characteristic polynomial where a copy of same rooted graph $G$ is attached by the root to \emph{each} vertex of $B$. Moreover, we establish if $(H_1,B_1)$ and $(H_2,B_2)$ are two coalescent pairs which are cospectral for any possible rooted graph $G$, then $(H_1,V(H_1)\setminus B_1)$ and $(H_2,V(H_2)\setminus B_2)$ will also always be cospectral for any possible rooted graph $G$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
