Extremal values for ratios of distances in trees

Barefoot, C. A.; Entringer, R. C.; Székely, László Á.

doi:10.1016/s0166-218x(97)00068-1

Cited by 38 publications

(29 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Part (i) improves on a result in [2], which states that, if T is a MAD tree of a given connected graph G, then there exists a vertex c in T such that every path in T starting at c is induced in G. We now prove that c can be chosen to be a median vertex of T .…”

Section: Structure Of Mad Trees In Interval Graphsmentioning

confidence: 67%

“…, v 4 . An interval representation of this graph is as follows: v 1 : [0, 1], v 2 : [0, 3], v 3 : [2,5], v 4 : [4,5], v 5 : [0, 5]. This graph has a unique MAD tree, the path v 1 , v 2 , v 3 , v 4 , v 5 .…”

Section: Discussionmentioning

confidence: 99%

“…Let T 1 and T 2 be the connected components of T − v j −1 v j containing v j −1 and v j , respectively. It is known (see [2]) that d T (c) d T (v i ) < d T (v j −1 ). Since the vertices in T 2 are further away (in T ) from v i , than from v j −1 , we obtain…”

Section: Lemma 1 (I)mentioning

confidence: 99%

See 2 more Smart Citations

A linear-time algorithm to compute a MAD tree of an interval graph

Dahlhaus

Dankelmann

Ravi

2004

Information Processing Letters

View full text Add to dashboard Cite

Section: Structure Of Mad Trees In Interval Graphsmentioning

confidence: 67%

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

A linear-time algorithm to compute a MAD tree of an interval graph

Dahlhaus

Dankelmann

Ravi

2004

Information Processing Letters

View full text Add to dashboard Cite

“…Two properties of the ordinary median, the second one due to [2], also hold for the c-median if c is strictly positive. The proofs for the ordinary median can be easily adapted to the c-median, so we omit them.…”

Section: A Useful Constructionmentioning

confidence: 98%

Average distance and generalised packing in graphs

Dankelmann

2010

Discrete Mathematics

View full text Add to dashboard Cite

a b s t r a c tLet G be a connected finite graph. The average distance µ(G) of G is the average of the distances between all pairs of vertices of G. For a positive integer k, a k-packing of G is a subset S of the vertex set of G such that the distance between any two vertices in S is greater than k. The k-packing number β k (G) of G is the maximum cardinality of a k-packing of G. We prove upper bounds on the average distance in terms of β k (G) and show that for fixed k the bounds are, up to an additive constant, best possible. As a corollary, we obtain an upper bound on the average distance in terms of the k-domination number, the smallest cardinality of a set S of vertices of G such that every vertex of G is within distance k of some vertex of S.

show abstract

“…, and σ(T )/σ T (w) were determined for a tree of given order in [1]. As an effort to further verify the negative correlation between the distance problem and the subtree problem, the extremal values of…”

Section: Introductionmentioning

confidence: 99%

Some extremal ratios of the distance and subtree problems in binary trees

Wang

2019

Applied Mathematics and Computation

View full text Add to dashboard Cite

Among many topological indices of trees the sum of distances σ(T ) and the number of subtrees F (T ) have been a long standing pair of graph invariants that are well known for their negative correlation. That is, among various given classes of trees, the extremal structures maximizing one usually minimize the other, and vice versa. By introducing the "local" versions of these invariants, σ T (v) for the sum of distance from v to all other vertices and F T (v) for the number of subtrees containing v, extremal problems can be raised and studied for vertices within a tree. This leads to the concept of "middle parts" of a tree with respect to different indices. A challenging problem is to find extremal values of the ratios between graph indices and corresponding local functions at middle parts or leaves. This problem also provides new opportunities to further verify the the correlation between different indices such as σ(T ) and F (T ). Such extremal ratios, along with the extremal structures, were studied and compared for the distance and subtree problems for general trees (Barefoot, Entringer and Székely, Discrete Applied Math 80 (1997); Székely and Wang, Electronic Journal of Combinatorics 20 (2013); Székely and Wang, Discrete Mathematics 322 (2014)). In this paper this study is extended to binary trees, a class of trees with numerous practical applications in which the extremal ratio problems appear to be even more complicated. After justifying some basic properties on the distance and subtree problems in trees and binary trees, characterizations are provided for the extremal structures achieving two extremal ratios in binary trees of given order. The generalization of this work to k-ary trees is also briefly discussed. The findings are compared with the previous established extremal structures in general trees. Lastly some potential future work is mentioned.2010 Mathematics Subject Classification. 05C05, 05C30.

show abstract

Extremal values for ratios of distances in trees

Cited by 38 publications

References 9 publications

A linear-time algorithm to compute a MAD tree of an interval graph

A linear-time algorithm to compute a MAD tree of an interval graph

Average distance and generalised packing in graphs

Some extremal ratios of the distance and subtree problems in binary trees

Contact Info

Product

Resources

About