并运算下的零和逆实现与性质(P)
Zero-sum Inverse Realization and Property~(P) under Join Operations
AI总结：
该研究提出图性质(P)的零和逆实现概念，证明阶数≥3且具性质(P)的图均有该实现，还证明性质(P)在图的并运算下保持，进而得到余图和字典积具性质(P)的充分条件。
AI中文摘要：
我们引入图G的性质(P)的零和逆实现，该实现通过在实现性质(P)的矩阵A∈S(G)上施加附加条件1ᵀA⁻¹1=0得到，其中1是全1向量。我们证明每个阶数至少为3且具有性质(P)的图都允许这样的零和逆实现。作为应用，我们证明性质(P)在两个阶数至少为3的图的并运算下保持，更一般地，在任意H下的一族阶数至少为3的图的H-并运算下也保持。由此，我们得到余图和图的字典积具有性质(P)的充分条件。全文给出了多个示例。
英文摘要：
We introduce zero-sum inverse realization of property (P) of a graph $G$, obtained by imposing an additional condition \( \mathbf{1}^{\top}A^{-1}\mathbf{1}=0, \) on a matrix $A\in S(G)$ realizing property (P), where $\mathbf{1}$ is the all-ones vector. We prove that every graph of order at least three having property (P) admits such a zero-sum inverse realization. As applications, we prove that property~(P) is preserved under the join of two graphs of order at least $3$ and, more generally, under the $H$-join of a family of graphs of order at least $3$, where $H$ is arbitrary. Consequently, we obtain sufficient conditions for cographs and lexicographic products of graphs to possess property~(P). Throughout the paper, many examples are given.
