HL-index of a graph

Abstract

Let G be a simple, connected graph with n vertices and eigenvalues λ₁ > λ₂ ≥ … ≥ 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.