The Fifth Coefficient of Adjoint Polynomial and a New Invariant

Abstract

Two graphs are defined to be adjointly equivalent if their complements are chromatically equivalent. In \([2, 7]\), Liu and Dong et al. give the first four coefficients \(b_0\), \(b_1\), \(b_2\), \(b_3\) of the adjoint polynomial and two invariants \(R_1\), \(R_2\), which are useful in determining the chromaticity of graphs. In this paper, we give the expression of the fifth coefficient \(b_4\), which brings about a new invariant \(R_3\). Using these new tools and the properties of the adjoint polynomials, we determine the chromatic equivalence class of \(\overline{B_{n-9,1,5}}\).