On the nullity of graphs with pendent vertices

Abstract

The nullity of a graph G, denoted by η ( G ) , is the multiplicity of the eigenvalue zero in its spectrum. Cheng and Liu [B. Cheng, B. Liu, On the nullity of graphs, Electron. J. Linear Algebra 16 (2007) 60–67] characterized the extremal graphs attaining the upper bound n - 2 and the second upper bound n - 3 . In this paper, as the continuance of it, we determine the extremal graphs with pendent vertices achieving the third upper bound n - 4 and fourth upper bound n - 5 . We then proceed recursively to construct all graphs with pendent vertices which satisfy η ( G ) > 0 . Our results provide a unified approach to determine n-vertex unicyclic (respectively, bicyclic and tricyclic) graphs which achieve the maximal and second maximal nullity and characterize n-vertex extremal trees attaining the second and third maximal nullity. As a consequence we, respectively, determine the nullity sets of trees, unicyclic graphs, bicyclic graphs and tricyclic graphs on n vertices.