Extremal values on the eccentric distance sum of trees

Geng, Xianya; Li, Shuchao; Zhang, Meng

doi:10.1016/j.dam.2013.05.023

Cited by 56 publications

(12 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since it has been proved in [10] that γ (P n ) = n/3 , combing with Lemma 2.4, we obtain the following theorem. Proof In view of Theorem 4.3, our result holds for n = 4, 5, 6.…”

Section: Theorem 42 Let T ∈ T Nγ Then H(t) ≥ T(n γ ) the Equalimentioning

confidence: 88%

Extremal values on the harmonic number of trees

Fan

Zhao

2014

International Journal of Computer Mathematics

Self Cite

View full text Add to dashboard Cite

Let G = (V (G), E(G)) be a simple connected graph. The harmonic number of G, denoted by H(G), is defined as the sum of the weights 2/(d(u) + d(v)) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this paper, some extremal problems on the harmonic number of trees are studied. The extremal values on the harmonic number of trees with given graphic parameters, such as pendant number, matching number, domination number and diameter, are determined. The corresponding extremal graphs are characterized, respectively.

show abstract

“…Since it has been proved in [10] that γ (P n ) = n/3 , combing with Lemma 2.4, we obtain the following theorem. Proof In view of Theorem 4.3, our result holds for n = 4, 5, 6.…”

Section: Theorem 42 Let T ∈ T Nγ Then H(t) ≥ T(n γ ) the Equalimentioning

confidence: 88%

Extremal values on the harmonic number of trees

Fan

Zhao

2014

International Journal of Computer Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Recently, mathematical properties of the eccentric distance sum of graphs have been studied. Mukungunugwa and Mukwembi [30] obtained the asymptotically sharp upper bounds on ξ d (G) with respect to the order and minimal degree of G. Geng, Zhang and one of the present authors [8] studied the relationship between ξ d and some other parameters, such as domination number, pendants and so on of trees. For more results on ξ d (G), one may be referred to [17,24,25,28] and references therein.…”

Section: Introductionmentioning

confidence: 86%

On the extremal total reciprocal edge-eccentricity of trees

Zhao

2016

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

The total reciprocal edge-eccentricity is a novel graph invariant with vast potential in structure activity/property relationships. This graph invariant displays high discriminating power with respect to both biological activity and physical properties. If $G=(V_G,E_G)$ is a simple connected graph, then the total reciprocal edge-eccentricity (REE) of $G$ is defined as $\xi^{ee}(G)=\sum_{uv\in E_G}(1/\varepsilon_G(u)+1/\varepsilon_G(v))$, where $\varepsilon_G(v)$ is the eccentricity of the vertex $v$. In this paper we first introduced four edge-grafting transformations to study the mathematical properties of the reciprocal edge-eccentricity of $G$. Using these elegant mathematical properties, we characterize the extremal graphs among $n$-vertex trees with given graphic parameters, such as pendants, matching number, domination number, diameter, vertex bipartition, et al. Some sharp bounds on the reciprocal edge-eccentricity of trees are determined.Comment: 16pages, 8figures, Journal of Mathematics Analysis and Applications (2015

show abstract

“…In view of [6,19,20,25], this distance-based graph invariant is well-studied for trees. It is natural and interesting to study the mathematical properties of this novel graph invariant on connected bipartite graphs.…”

Section: Introductionmentioning

confidence: 99%

“…This work led to finding the trees of fixed order with third and fourth minimal eccentric distance sums. Geng et al [6] characterized the tree among n-vertex trees with domination number γ having the minimal EDS and the graphs among n-vertex trees with domination number γ satisfying n = kγ having the maximal EDS is identified, respectively, for k = 2, 3, n/3, n/2. Sharp upper and lower bounds on the EDS among the n-vertex trees with k leaves are determined in [6], whereas the trees among the n-vertex trees with a given bipartition having the minimal, second minimal and third minimal eccentric distance sums are, respectively, determined as well in [6].…”

Section: Introductionmentioning

confidence: 99%

“…Geng et al [6] characterized the tree among n-vertex trees with domination number γ having the minimal EDS and the graphs among n-vertex trees with domination number γ satisfying n = kγ having the maximal EDS is identified, respectively, for k = 2, 3, n/3, n/2. Sharp upper and lower bounds on the EDS among the n-vertex trees with k leaves are determined in [6], whereas the trees among the n-vertex trees with a given bipartition having the minimal, second minimal and third minimal eccentric distance sums are, respectively, determined as well in [6]. Azari and Iranmanesh [1] presented explicit formulas for computing the EDS of the some graph operations such as the Cartesian product, join, composition, disjunction, symmetric difference, cluster and corona product of graphs.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation