On the core-nilpotent decomposition of threshold graphs

In this work, we show that the core-nilpotent decomposition of the adjacency matrix \(A(G)\) of a threshold graph \(G\) can be obtained directly from the graph through its null decomposition. More precisely, we construct two invertible matrices \(Q\) and \(C\), defined in terms of the adjacency relations of \(G\), such that $Q^{-1}A(G)Q$ is a block diagonal matrix with two blocks. The first block is \(C\), an \(r\times r\) matrix satisfying $\operatorname{rk}(C)=\operatorname{rk}(A(G))=r$, and the second block is the zero matrix. Consequently, the core-nilpotent decomposition of \(A(G)\) can be read directly from the combinatorial structure of \(G\).