2013
DOI: 10.1155/2013/906756
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Wiener Index of Graphs with Radius Two

Abstract: The Wiener index of a graph is the sum of the distances between all pairs of vertices. It has been one of main descriptors that correlate a chemical compound's molecular graph with experimentally gathered data regarding the compound's characteristics. We characterize graphs with the maximum Wiener index among all graphs of order . with radius two. In addition, we pose a conjecture concerning the minimum Wiener index of graphs with given radius. If this conjecture is true, it will be able to answer an open ques… Show more

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Cited by 8 publications
(15 citation statements)
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“…Regarding this problem, Chen et al [5] stated the following conjecture. For integers n, r, and s with n ≥ 2r, r ≥ 3, and n − 2r + 1 ≥ s ≥ 1, construct a graph G n,r,s from a 2r-cycle v 1 v 2 · · · v 2r so that v 1 is replaced by K s and v 2 is replaced by K n−2r+2−s , connect v 2r to each vertex of K s , connect each vertex of K s to each vertex of K n−2r+2−s , and connect each vertex of K n−2r+2−s to v 3 (in other words v 1 is replicated s−1 times, and v 2 is replicated n−2r +1−s times).…”
Section: Graphs With Prescribed Diameter/radiusmentioning
confidence: 99%
“…Regarding this problem, Chen et al [5] stated the following conjecture. For integers n, r, and s with n ≥ 2r, r ≥ 3, and n − 2r + 1 ≥ s ≥ 1, construct a graph G n,r,s from a 2r-cycle v 1 v 2 · · · v 2r so that v 1 is replaced by K s and v 2 is replaced by K n−2r+2−s , connect v 2r to each vertex of K s , connect each vertex of K s to each vertex of K n−2r+2−s , and connect each vertex of K n−2r+2−s to v 3 (in other words v 1 is replicated s−1 times, and v 2 is replicated n−2r +1−s times).…”
Section: Graphs With Prescribed Diameter/radiusmentioning
confidence: 99%
“…The question of determining the minimum total distance among all graphs or digraphs of order n and (out)radius r and characterizing the extremal (di)graphs has been solved for n large enough compared with r. This partially solves the conjecture of Chen et al [4]. It might be very challenging to solve the question completely for every order, as there might be sporadic extremal graphs other than Q3 which are also different from those conjectured in the initial conjecture.…”
Section: Resultsmentioning
confidence: 98%
“…The question of determining the minimum total distance among all graphs or digraphs of order n and (out)radius r and characterizing the extremal (di)graphs has been solved for n large enough compared with r. This partially solves the conjecture of Chen, Wu and An [4].…”
Section: Discussionmentioning
confidence: 99%
See 1 more Smart Citation
“…Here there are 6 non-isomorphic extremal graphs (a = 1 gives twice the same graph). Chen, Wu and An [6] had conjectured that the graphs G n,r,s minimize the total distance among all graphs of order n and radius r. This was proven for n sufficiently large compared with r in [4]. It is natural to think there is a relation between maximizing the size and minimizing the total distance.…”
Section: Introductionmentioning
confidence: 99%