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On the harmonic index of bicyclic graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 2، دوره 3، شماره 2، اسفند 2018، صفحه 121-142 اصل مقاله (341.28 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2018.26171.1081 | ||
| نویسنده | ||
| Reza Rasi* | ||
| Azarbaijan Shahid Madani University | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| harmonic index؛ bicyclic graphs؛ trees | ||
| مراجع | ||
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