HL-index of a graph

Let G be a simple, connected graph with n vertices and eigenvalues λ 1 > λ 2 ≥ … ≥ λ n . If n is even, define H = n /2 and L = H + 1. If n is odd, define H = L = ( n + 1)/2. Define the HL-index of G to be R ( G ) = max(| λ H |, | λ L |). The eigenvalues λ H and λ L appear in chemical graph theory in the study of molecular stability. In this paper, bounds on HL-index for chemical and general graphs are studied. It is shown that there exist graphs with arbitrarily large HL-index.