Mean distance and minimum degree

Kouider, Mekkia; Winkler, Peter

doi:10.1002/(sici)1097-0118(199705)25:1<95::aid-jgt7>3.0.co;2-d

Cited by 42 publications

(16 citation statements)

References 5 publications

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“…[1][2][3][4][5][6][7][8][9]. This distance is used to study the central concepts like center, median, and centroid of a graph [10][11][12][13][14][15][16][17][18][19][20][21][22]. With regard to convexity, this distance is the basis of some geodetic parameters such as geodetic number, connected geodetic number, upper geodetic number and forcing geodetic number [23][24][25][26][27][28][29][30][31][32].…”

Section: Introductionmentioning

confidence: 99%

Monophonic Distance in Graphs

Santhakumaran

Titus

2011

Discrete Math. Algorithm. Appl.

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For any two vertices u and v in a connected graph G, a u – v path is a monophonic path if it contains no chords, and the monophonic distance dm(u, v) from u to v is defined as the length of a longest u – v monophonic path in G. A u – v monophonic path of length dm(u, v) is called a u – v monophonic. The monophonic eccentricity em(v) of a vertex v in G is the maximum monophonic distance from v to a vertex of G. The monophonic radius rad m G of G is the minimum monophonic eccentricity among the vertices of G, while the monophonic diameter diam m G of G is the maximum monophonic eccentricity among the vertices of G. It is shown that rad m G ≤ diam m G for every connected graph G and that every pair a, b of positive integers with a ≤ b is realizable as the monophonic radius and monophonic diameter of some connected graph. Also, for any three positive integers a, b and c with 3 ≤ a ≤ b ≤ c, there is a connected graph G such that rad G = a, rad m G = b and rad DG = c; and for any three positive integers a, b and c with 5 ≤ a ≤ b ≤ c, there is a connected graph G such that diam G = a, diam m G = b and diam D G = c, where rad G, diam G, rad DG and diam D G denote the radius, diameter, detour radius and detour diameter, respectively. The monophonic center of G is the subgraph induced by the vertices of G having monophonic eccentricity rad m G and it is shown that every graph is the monophonic center of some connected graph and also that the monophonic center Cm(G) of every connected graph G is a subgraph of some block of G.

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Section: Introductionmentioning

confidence: 99%

Monophonic Distance in Graphs

Santhakumaran

Titus

2011

Discrete Math. Algorithm. Appl.

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“…Hence e ≤ 2n−8 5 . A simple differentiation shows that (2) is maximized, subject to the constraint e ≤ 2n− 8 5 , for e = 2n−8…”

Section: Results On 5-edge-connected Graphsmentioning

confidence: 99%

“…Let G = (V, E) be a connected graph of order n. The average distance µ(G) of G is defined as µ(G) = n 2 −1 {x,y}⊂V d G (x, y), where d G (x, y) denotes the distance between the vertices x and y in G. The average distance has been investigated by several authors and under various names; for instance, Kouider and Winkler [8], Favaron, Kouider, Mahéo [6], Doyle and Graver [4] depict average distance as the mean distance; Wuchty and Stadler [16] use the term average path length; Bermond, Liu and Syska [1] use the term mean eccetricity; whilst Plesník [12], Soltés [14], work with the total distance of a graph and call it the transmission. The transmission differs from the average distance by only a factor of n(n−1) .…”

Section: Introductionmentioning

confidence: 99%

Average Distance and Edge-Connectivity I

Dankelmann¹,

Mukwembi²,

Swart³

2008

SIAM J. Discrete Math.

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The average distance µ(G) of a connected graph G of order n is the average of the distances between all pairs of vertices of G. We prove that if G is a λ-edge-connected graph of order n, then the bounds µ(G) ≤ 2n/15 + 9, if λ = 5, 6, µ(G) ≤ n/9 + 10, if λ = 7 and µ(G) ≤ n/(λ + 1) + 5, if λ ≥ 8 hold. Our bounds are shown to be best possible and our results solve a problem of Plesník [12].

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“…One of the oldest results on upper bounds of this quantity is that amongst all connected graphs of given order, the path has the maximum Wiener index. For graphs of given minimum degree, this result was improved independently by several authors, among them Kouider and Winkler [10] and Dankelmann and Entringer [2], who proved the following bound. T 1.3 [2,10].…”

Section: Introductionmentioning

confidence: 91%