The singular acyclic matrices of even order with a P-set of maximum size

Let $m_A(\lambda)$ denote the multiplicity of the eigenvalue $\lambda$ of a given $n$-by-$n$ symmetric matrix $A$. Set $A(\alpha)$ for the principal submatrix of $A$ obtained after deleting the rows and columns indexed by the nonempty subset $\alpha$ of $\{1,\ldots,n\}$. When $m_{A(\alpha)}(0)=m_{A}(0)+|\alpha|$, we call $\alpha$ a P-set of $A$. The maximum size of a P-set of $A$ is denoted by $P_s(A)$. It is known that $P_{s}(A)\leqslant\left\lfloor\frac{n}{2}\right\rfloor$ and this bound is not sharp for singular acyclic matrices of even order. In this paper, we find the bound for this case and classify all of the underlying trees. Some illustrative examples are provided.