Average distance and independence number

Dankelmann, Peter

doi:10.1016/0166-218x(94)90095-7

Cited by 60 publications

(43 citation statements)

References 2 publications

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“…Clearly, the second conjecture is stronger than the first, which was proved by Chung [2]. An alternate proof of the first conjecture was proposed by Dankelmann [3], who also described the graphs G maximizing the average distance for a given order and a given value of α (G).…”

Section: ): µ(G) ≤ α(G) (Wow 2) and µ(G) ≤ α 2 (G)/2 (Wow 747)mentioning

confidence: 95%

“…Indeed, G is connected since the complement of a disconnected forest is connected, and F (G) is at most 3 (otherwise G would contain aK 3 , a P 4 , or a 2K 2 , implying thatḠ contains a triangle, a P 4 or a square).…”

Section: Moreover If F (G) Is At Least 4 G Is (N F (G))-forest-extmentioning

confidence: 98%

“…We use the same notation as Dankelmann [3] for several quantities. For instance, given a graph G and a vertex x of G, σ G (x) denotes the transmission of x in G, that is,…”

Section: ): µ(G) ≤ α(G) (Wow 2) and µ(G) ≤ α 2 (G)/2 (Wow 747)mentioning

confidence: 99%

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Average distance and maximum induced forest

Hansen¹,

Hertz²,

Marcotte³

et al. 2008

Journal of Graph Theory

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Abstract:With the help of the Graffiti system, Fajtlowicz conjectured around 1992 that the average distance between two vertices of a connected graph G is at most half the maximum order of an induced bipartite subgraph of G, denoted α 2 (G). We prove a strengthening of this conjecture by showing that the average distance between two vertices of a connected graph G is at most half the maximum order of an induced forest, denoted F(G). Moreover, we characterize the graphs maximizing the average distance among all graphs G having a fixed number of vertices and a fixed value of F(G) or α 2 (G). Finally, we conjecture that the average distance between two vertices of a connected graph is at most half the maximum order of an induced linear forest (where a linear forest is a union of paths).

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Section: ): µ(G) ≤ α(G) (Wow 2) and µ(G) ≤ α 2 (G)/2 (Wow 747)mentioning

confidence: 95%

Section: Moreover If F (G) Is At Least 4 G Is (N F (G))-forest-extmentioning

confidence: 98%

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Average distance and maximum induced forest

Hansen¹,

Hertz²,

Marcotte³

et al. 2008

Journal of Graph Theory

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“…Zhang et al [11] and Wang [12] also investigated the trees with the maximum Wiener index among all trees with a given degree sequence. Dankelmann [13] characterized the trees having the maximum Wiener index in (n, i). Du and Zhou [14] identified the trees with the minimum Wiener index in (n, i).…”

mentioning

confidence: 99%

Ordering trees with given matching number by their Wiener indices

Tan

Wang

et al. 2014

J. Appl. Math. Comput.

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The Wiener index of a connected graph is the sum of distances between all pairs of vertices in the graph. Let (n, i) be the set of all trees with order n and matching number i. In this article, we give five graphic transformations that change the Wiener index of graphs, then with them we determine the second to sixth trees in (2i + 1, i) and the third to eighth trees in (n, i) for n ≥ 2i + 2 having the smallest Wiener indices.

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“…If we restrict our self to the graph theory, there exists a connection between average distance and independence number [3,4], domination number [5], generalized packing [6], to name just a few. Hence there also exists a connection between the Wiener index and mentioned graph concepts.…”

mentioning

confidence: 99%

Wiener index of strong product of graphs

Peterin¹,

Pleteršek²

2018

OpMath

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Abstract. The Wiener index of a connected graph G is the sum of distances between all pairs of vertices of G. The strong product is one of the four most investigated graph products. In this paper the general formula for the Wiener index of the strong product of connected graphs is given. The formula can be simplified if both factors are graphs with the constant eccentricity. Consequently, closed formulas for the Wiener index of the strong product of a connected graph G of constant eccentricity with a cycle are derived.

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Average distance and independence number

Cited by 60 publications

References 2 publications

Average distance and maximum induced forest

Average distance and maximum induced forest

Ordering trees with given matching number by their Wiener indices

Wiener index of strong product of graphs

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