On singular line graphs of trees

Abstract

This paper studies line graphs of trees. We show that the nullity of a singular line graph L T of a tree T is one and that the order of T is even. A nut graph is a singular graph (≠K 1 ) of nullity one such that a kernel eigenvector (an eigenvector in the nullspace of its adjacency matrix) has no zero entries. We show that, if L T has a terminal K 3 , then it is either a nut graph, or else, there exist cut vertices such that at least one component of the graph obtained on deleting them, is a nut graph. Moreover, if L T is singular and not a nut graph, then at least one spectrum, in the deck of spectra of the one-vertex-deleted subgraphs of L T , has the zero eigenvalue repeated.