On the harmonic index of bicyclic graphs

Abstract

The harmonic index of a graph $G$, denoted by $H(G)$, is defined asthe sum of weights $2/[d(u)+d(v)]$ over all edges $uv$ of $G$, where$d(u)$ denotes the degree of a vertex $u$. Hu and Zhou [Y. Hu and X. Zhou, WSEAS Trans. Math. {bf 12} (2013) 716--726] proved that for any bicyclic graph $G$ of order $ngeq 4$, $H(G)le frac{n}{2}-frac{1}{15}$ and characterize all extremal bicyclic graphs.In this paper, we prove that for any bicyclic graph $G$ of order $ngeq 4$ and maximum degree $Delta$, $$H(G)le left{begin{array}{ll}frac{3n-1}{6} & {rm if}; Delta=4\&\2(frac{2Delta-n-3}{Delta+1}+frac{n-Delta+3}{Delta+2}+frac{1}{2}+frac{n-Delta-1}{3}) & {rm if};Deltage 5 ;{rm and}; nle 2Delta-4\&\2(frac{Delta}{Delta+2}+frac{Delta-4}{3}+frac{n-2Delta+4}{4}) & {rm if};Deltage 5 ;{rm and};nge 2Delta-3,\end{array}right.$$ and characterize all extreme bicyclic graphs.