Extremal values on the eccentric distance sum of trees
Abstract
Let $G=(V_G, E_G)$ be a simple connected graph. The eccentric distance sum of $G$ is defined as $\xi^{d}(G) = \sum_{v\in V_G}\varepsilon_{G}(v)D_{G}(v)$, where $\varepsilon_G(v)$ is the eccentricity of the vertex $v$ and $D_G(v) = \sum_{u\in V_G}d_G(u,v)$ is the sum of all distances from the vertex $v$. In this paper the tree among $n$-vertex trees with domination number $\gamma$ having the minimal eccentric distance sum is determined and the tree among $n$-vertex trees with domination number $\gamma$ satisfying $n = k\gamma$ having the maximal eccentric distance sum is identified, respectively, for $k=2,3,\frac{n}{3},\frac{n}{2}$. Sharp upper and lower bounds on the eccentric distance sums among the $n$-vertex trees with $k$ leaves are determined. Finally, the trees among the $n$-vertex trees with a given bipartition having the minimal, second minimal and third minimal eccentric distance sums are determined, respectively.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2012
- DOI:
- 10.48550/arXiv.1207.0083
- arXiv:
- arXiv:1207.0083
- Bibcode:
- 2012arXiv1207.0083L
- Keywords:
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- Mathematics - Combinatorics;
- 05C50;
- 15A18
- E-Print:
- 15 Pages, 8 figures