Zero-sum Inverse Realization and Property~(P) under Join Operations

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.