Spanning Trees with Many Leaves and Average Distance

“…Upper bounds on the Gutman index of a graph of given order and diameter were studied in [4,9]. In [9] it was proved, that Gut(G) ≤ (1/16)d(n − d) 4 + O(n 4 ) and consequently Gut(G) ≤ (2 4 /5 5 )n 5 + O(n 4 ).…”

Section: Introductionmentioning

confidence: 99%

“…It is not difficult to show that the extremal tree, which has the minimum Wiener index among trees of order n and diameter d, is the path of length d (containing d + 1 vertices) with the central vertex joined to the other n − d − 1 vertices; see [12]. The problem of finding an upper bound on the Wiener index of a tree (or graph) in terms of order and diameter is quite challenging; it was addressed by Plesník [10] in 1975, and restated by DeLaViña and Waller [5], but still remains unresolved to this date. In this paper, we give a starting point to solving this long-standing problem.…”

Section: Introductionmentioning

confidence: 99%

Wiener Index of Trees of Given Order and Diameter at Most

Mukwembi¹,

Vetrík

2013

Bull. Aust. Math. Soc.

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The long-standing open problem of finding an upper bound for the Wiener index of a graph in terms of its order and diameter is addressed. Sharp upper bounds are presented for the Wiener index, and the related degree distance and Gutman index, for trees of order n and diameter at most 6.2010 Mathematics subject classification: primary 05C12; secondary 05C05, 05C07. Keywords and phrases: Wiener index, degree distance, Gutman index, tree, diameter. IntroductionLet G be a graph with vertex set V(G) and order n. We denote the distance between two vertices u, v in G by d G (u, v) (or simply d(u, v)); the diameter of G will be denoted by d(G) (or d), the eccentricity of a vertex v will be denoted by ec(v) and the degree of v will be denoted by deg (v). Let N G i (v) (or simply N i (v)) be the set of vertices at distance i from v in G. Let u, v be two adjacent (nonadjacent) vertices of a graph G. Then G = G − uv (G = G + uv) is obtained by removing the edge uv from G (by adding the edge uv to G).The Wiener index is the oldest topological index. It has been investigated in the mathematical, chemical and computer science literature since the 1940s. The Wiener index W(G) of a connected graph G is defined as the sum of the distances between all unordered pairs of vertices. The minimum value of the Wiener index of a graph (of a tree) of given order is attained by the complete graph (by the star), and the maximum value is attained by the path.The degree distance, a variant of the Wiener index, is defined as and the Gutman index is defined asThe smallest value of the degree distance and Gutman index of graphs of order n is attained by stars (see [1,11]). Turning to upper bounds on the degree distance, in 1999 Tomescu [11] conjectured the asymptotic upper bound D (G) ≤ (1/27)n 4 + O(n 3 ). Nine years later, Bucicovschi and Cioabǎ [2] commented that Tomescu's conjecture 'seems difficult at present time'. In the following year Dankelmann et al. [3] considered this problem and though they came close to proving the conjecture, their proof was inadequate to meet the O(n 3 ) error term. Recently, Morgan et al., in a submitted paper ('On a conjecture by Tomescu'), salvaged enough from the proof given in [3] and solved Tomescu's conjecture completely. There one can also find upper bounds on the degree distance of graphs of given order and diameter. Upper bounds on the Gutman index of a graph of given order and diameter were studied in [4,9]. In [9] it was proved, that Gut(. In this paper we study the indices mentioned above for trees of given order and diameter. Since Klein et al. [7] showed that for every tree T of order n,and in [6] Gutman proved thatany result on W(T ) yields a similar result on D (T ) and Gut(T ). It is not difficult to show that the extremal tree, which has the minimum Wiener index among trees of order n and diameter d, is the path of length d (containing d + 1 vertices) with the central vertex joined to the other n − d − 1 vertices; see [12]. The problem of finding an upper bound on the Wiener index of a ...

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“…The computer programs Graffiti [5] and AutoGraphiX [1] with the classical 1984 paper by Plesnik [8] are three of the best sources for problems and conjectures related to average distance and total distance (Wiener index). These sources contain some pretty and long-standing problems on this topic (see also [4] and the references cited therein).…”

Section: Introductionmentioning

confidence: 99%

On the Total Distance and Diameter of Graphs

Hua¹

2018

Bull. Aust. Math. Soc.

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The total distance (or Wiener index) of a connected graph $G$ is the sum of all distances between unordered pairs of vertices of $G$. DeLaViña and Waller [‘Spanning trees with many leaves and average distance’, Electron. J. Combin.15(1) (2008), R33, 14 pp.] conjectured in 2008 that if $G$ has diameter $D>2$ and order $2D+1$, then the total distance of $G$ is at most the total distance of the cycle of the same order. In this note, we prove that this conjecture is true for 2-connected graphs.

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Spanning Trees with Many Leaves and Average Distance

Cited by 30 publications

References 23 publications

Mathematical aspects of Wiener index

Mathematical aspects of Wiener index

Wiener Index of Trees of Given Order and Diameter at Most

On the Total Distance and Diameter of Graphs

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