On (k + 1)-line graphs of k-trees and their nullities

The nullity of a graph, denoted by η(G), is the multiplicity of zero as an eigenvalue of the adjacency matrix of G. Fiorini, Gutman and Sciriha (2005) in [5] presented an upper bound for the nullities of trees in terms of the order n and the maximum degree Δ. In our work we show that, under the same conditions, all possible nullities below that bound are attained. This result allows us to obtain an upper bound for the nullities of the (k+1)-line graphs of a particular family of k-trees, generalizing a known result about the nullities of line graphs of trees. We also present a general characterization for the (k+1)-line graphs of k-trees.