@article {
author = {Rasi, Reza},
title = {On the harmonic index of bicyclic graphs},
journal = {Communications in Combinatorics and Optimization},
volume = {3},
number = {2},
pages = {121-142},
year = {2018},
publisher = {Azarbaijan Shahid Madani University},
issn = {2538-2128},
eissn = {2538-2136},
doi = {10.22049/cco.2018.26171.1081},
abstract = {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 $n\geq 4$ and maximum degree $\Delta$, $$H(G)\le \left\{\begin{array}{ll}\frac{3n-1}{6} & {\rm if}\; \Delta=4\\&\\2(\frac{2\Delta-n-3}{\Delta+1}+\frac{n-\Delta+3}{\Delta+2}+\frac{1}{2}+\frac{n-\Delta-1}{3}) & {\rm if}\;\Delta\ge 5 \;{\rm and}\; n\le 2\Delta-4\\&\\2(\frac{\Delta}{\Delta+2}+\frac{\Delta-4}{3}+\frac{n-2\Delta+4}{4}) & {\rm if}\;\Delta\ge 5 \;{\rm and}\;n\ge 2\Delta-3,\\\end{array}\right.$$ and characterize all extreme bicyclic graphs.},
keywords = {harmonic index,bicyclic graphs,trees},
url = {http://comb-opt.azaruniv.ac.ir/article_13746.html},
eprint = {http://comb-opt.azaruniv.ac.ir/article_13746_f0c613a9e6610951d57150aad863731f.pdf}
}