On the core-nilpotent decomposition of unicyclic graphs

Abstract

In this work we show, through the null decomposition of unicyclic graphs given in cite{Maikon3}, that the core-nilpotent decomposition of (A(U)), the adjacency matrix of a unicyclic graph (U), can be obtained directly from the graph itself. In other words, we give two invertible matrices (Q) and (C), expressed in terms of the adjacency relations of (U), such that (Q^{-1}A(U)Q) is a block diagonal matrix with two blocks. The first block is (C) (an (rimes r) matrix such that (ank{K} = ank{A(U)} = r)), and the second block is a zero matrix.