The positive and the negative inertia index of line graphs of trees

The positive (resp., negative) inertia index of G  , denoted by p(G)p(G) (resp., n(G)n(G)), is defined to be the number of positive (resp., negative) eigenvalues of its adjacency matrix. The nullity of G  , denoted by η(G)η(G), is defined to be the multiplicity of the eigenvalue zero in the adjacency spectrum of G. In 2001, Gutman and Sciriha [11] proved that the nullity of the line graph of a tree is either 0 or 1. In this paper we consider the positive and the negative inertia index for the line graph LTLT of a tree T  , showing that ε(T)+12⩽p(LT)⩽ε(T)+1, where ε(T)ε(T) denotes the number of internal (non-pendant) edges contained in T  . The extremal trees for which p(LT)p(LT) attains the upper bound and the lower bound are respectively characterized. It is shown that LTLT is nonsingular if p(LT)p(LT) attains the upper bound and LTLT is singular if p(LT)p(LT) attains the lower bound.