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Graphs with fixed number of pendent vertices and minimal first Zagreb index | ||
| Transactions on Combinatorics | ||
| مقاله 4، دوره 4، شماره 1، خرداد 2015، صفحه 43-48 اصل مقاله (210.65 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2015.6029 | ||
| نویسندگان | ||
| Ivan Gutman* 1؛ Muhammad Kamran Jamil2؛ Naveed Akhter2 | ||
| 1University of Kragujevac Kragujevac, Serbia | ||
| 2Government College University | ||
| چکیده | ||
| The first Zagreb index $M_1$ of a graph $G$ is equal to the sum of squares of degrees of the vertices of $G$. Goubko proved that for trees with $n_1$ pendent vertices, $M_1 \geq 9\,n_1-16$. We show how this result can be extended to hold for any connected graph with cyclomatic number $\gamma \geq 0$. In addition, graphs with $n$ vertices, $n_1$ pendent vertices, cyclomatic number $\gamma$, and minimal $M_1$ are characterized. Explicit expressions for minimal $M_1$ are given for $\gamma=0,1,2$, which directly can be extended for $\gamma>2$. | ||
| کلیدواژه‌ها | ||
| degree (of vertex)؛ Zagreb index؛ First Zagreb index؛ extremal graphs | ||
| مراجع | ||
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