Rasi, R. (2018). On the harmonic index of bicyclic graphs. Communications in Combinatorics and Optimization, 3(2), 121-142. doi: 10.22049/cco.2018.26171.1081

Reza Rasi. "On the harmonic index of bicyclic graphs". Communications in Combinatorics and Optimization, 3, 2, 2018, 121-142. doi: 10.22049/cco.2018.26171.1081

Rasi, R. (2018). 'On the harmonic index of bicyclic graphs', Communications in Combinatorics and Optimization, 3(2), pp. 121-142. doi: 10.22049/cco.2018.26171.1081

Rasi, R. On the harmonic index of bicyclic graphs. Communications in Combinatorics and Optimization, 2018; 3(2): 121-142. doi: 10.22049/cco.2018.26171.1081

The harmonic index of a graph $G$, denoted by $H(G)$, is defined as the 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 $Delta$, $$frac{1}{2} H(G)le left{begin{array}{ll} frac{3n-1}{6} & {rm if}; Delta=4\ &\ frac{2Delta-n-3}{Delta+1}+frac{n-Delta+3}{Delta+2}+frac{1}{2}+frac{n-Delta-1}{3} & {rm if};Deltage 5 ;{rm and}; nle 2Delta-4\ &\ frac{Delta}{Delta+2}+frac{Delta-4}{3}+frac{n-2Delta+4}{4} & {rm if};Deltage 5 ;{rm and};nge 2Delta-3,\ end{array}right.$$ and characterize all extreme bicyclic graphs.