Upper bounds on the sum of powers of the degrees of a simple planar graph |
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Authors: | Jochen Harant Stanislav Jendrol Tomá? Madaras |
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Institution: | 1. Institut Fur Mathematik, Tu Ilmenau Postfach 100565, D‐98684 Ilmenau, Germany;2. Institute of Mathematics, P. J. Safarik University Jesenna;3. 5, SK‐04154 Kosice Slovak Republic |
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Abstract: | For a simple planar graph G and a positive integer k, we prove the upper bound 2(n ? 1)k + 4k(n ? 4) + 2·3k ? 2((δ + 1)k ? δk)(3n ? 6 ? m) on the sum of the kth powers of the degrees of G, where n, m, and δ are the order, the size, and the minimum degree of G, respectively. The bound is tight for all m with 0?3n ? 6 ? m≤?n/2? ? 2 and δ = 3. We also present upper bounds in terms of order, minimum degree, and maximum degree of G. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:112‐123, 2011 |
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Keywords: | degree sum planar graph |
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