On the harmonic index of bicyclic graphs

Document Type : Original paper

Author

Azarbaijan Shahid Madani University

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

Main Subjects

References

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Communications in Combinatorics and Optimization
Volume 3, Issue 2
December 2018
Pages 121-142

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