Comment on “Complete solution to a conjecture on Randić index”

In this paper we will show that the proof of Theorem 2.1 from “Complete solution to a conjecture on Randi? index”, by Xueliang Li, Bolian Liu and Jianxi Liu, European Journal of Operational Research 200, Issue 1, (2010), 9–13, is not correct. They tried to prove the conjecture given by M. Aouchiche, P. Hansen in “On a conjecture about the Randi? index” (Discrete Mathematics, 307 (2), 2007, 262–265), but they failed in it. The mathematical model given by them is a problem of quadratic programming which they tried to solve by wrong use of linear programming. This error invalidates all further work.     

On a conjecture of the Randić index and the minimum degree of graphs

The Randi? index $R\left(G\right)$ of a graph $G$ is defined by $R\left(G\right)={\sum }_{uv}\frac{1}{\sqrt{d\left(u\right)d\left(v\right)}}$, where $d\left(u\right)$ is the degree of a vertex $u$ and the summation extends over all edges $uv$ of $G$. Delorme et al. (2002)  [6] put forward a conjecture concerning the minimum Randi? index among all$n$-vertex connected graphs with the minimum degree at least $k$. In this work, we show that the conjecture is true given the graph contains $k$ vertices of degree $n?1$. Further, it is true among $k$-trees.     

On a conjecture of the Randi? index

The Randi? index of a graph G is defined as , where d(u) is the degree of vertex u and the summation goes over all pairs of adjacent vertices u, v. A conjecture on R(G) for connected graph G is as follows: R(G)≥r(G)−1, where r(G) denotes the radius of G. We proved that the conjecture is true for biregular graphs, connected graphs with order n≤10 and tricyclic graphs.     

A quadratic programming approach to the Randić index

Let G(k, n) be the set of connected graphs without multiple edges or loops which have n vertices and the minimum degree of vertices is k. The Randi? index χ = χ(G) of a graph G   is defined by χ(G)=_∑(uv)(_δuδv)^-1/2$\chi (G)={\sum }_{(\mathit{uv})}({\delta }_u{\delta }_v{)}^{-1/2}$, where δ_u is the degree of vertex u and the summation extends over all edges (uv) of G. Caporossi et al. [G. Caporossi, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs IV: Chemical trees with extremal connectivity index, Computers and Chemistry 23 (1999) 469–477] proposed the use of linear programming as one of the tools for finding the extremal graphs. In this paper we introduce a new approach based on quadratic programming for finding the extremal graphs in G(k, n) for this index. We found the extremal graphs or gave good bounds for this index when the number n_k of vertices of degree k is between n − k and n. We also tried to find the graphs for which the Randi? index attained its minimum value with given k (k ? n/2) and n. We have solved this problem partially, that is, we have showed that the extremal graphs must have the number n_k of vertices of degree k less or equal n − k and the number of vertices of degree n − 1 less or equal k.     

The smallest Randić index for trees

The general Randi? index R _α (G) is the sum of the weight d(u)d(v) ^α over all edges uv of a graph G, where α is a real number and d(u) is the degree of the vertex u of G. In this paper, for any real number α?≠?0, the first three minimum general Randi? indices among trees are determined, and the corresponding extremal trees are characterized.     

Conjugated tricyclic graphs with the maximum zeroth-order general Randić index

Let G be a simple connected graph and α be a given real number. The zeroth-order general Randi? index of G is defined as ⁰ R _α (G)=∑ _v∈V(G)[d _G (v)] ^α , where d _G (v) denotes the degree of the vertex v of G. In this paper, for any α>2, we give sharp upper bounds of the zeroth-order general Randi? index ⁰ R _α of all conjugated tricyclic graphs with 2m vertices.     

Extremal trees with given degree sequence for the Randić index

The Randi? index of a graph G is the sum of $((d(u))(d(v)){)}^{\alpha }$ over all edges $\mathit{uv}$ of G, where $d(v)$ denotes the degree of $v$ in G, $\alpha \ne 0$. When $\alpha =1$, it is the weight of a graph. Delorme, Favaron, and Rautenbach characterized the trees with a given degree sequence with maximum weight, where the question of finding the tree that minimizes the weight is left open. In this note, we characterize the extremal trees with given degree sequence for the Randi? index, thus answering the same question for weight. We also provide an algorithm to construct such trees.     

Trees with second minimum general Randić index for α>0

The general Randi? index R _α (G) of a graph G is the sum of the weights (d(u)d(v)) ^α of all edges uv of G, where α is a real number(α≠0) and d(u) denotes the degree of the vertex u. We have known that P _n has minimum general Randi? index for α>0 among trees when n≥5. In this paper, we prove that P _n,3 has second minimum general Randi? index for α>0 among trees when n≥7.