Variable neighborhood search for extremal graphs: IV: Chemical trees with extremal connectivity index

doi:10.1016/S0097-8485(99)00031-5

Computers & Chemistry

Volume 23, Issue 5, 1 September 1999, Pages 469-477

https://doi.org/10.1016/S0097-8485(99)00031-5 Get rights and content

Abstract

By means of the variable neighborhood search algorithm, a newly designed heuristic approach to combinatorial optimization, we established the structure of the chemical trees possessing extremal (maximal and minimal) values for the Randić connectivity index (χ). These findings were eventually corroborated by rigorous mathematical proofs. As could have been anticipated, the n-vertex tree with maximum χ is the path. The n-vertex chemical tree with minimum χ-value is not unique. The structures of such chemical trees (which should be considered as the graph representations of the most branched alkanes) are fully characterized.

Introduction

The connectivity index χ, conceived by Milan Randić a quarter of century ago (Randić, 1975) and often referred to as the Randić index, is certainly the molecular-graph-based structure-descriptor that found the most numerous applications in organic chemistry, medicinal chemistry and pharmacology. Countless papers and two books (Kier and Hall, 1976, Kier and Hall, 1986) have been devoted to χ and its various generalizations; for recent research on the connectivity index see the article (Estrada et al., 1998) and the references quoted therein.

After a few early results (Gutman, 1985, Gutman, 1986), research on mathematical properties of the connectivity index was stimulated by a series of papers describing conjectures obtained with the automated system ‘Graffiti’ (Fajtlowicz, 1988a, Fajtlowicz, 1987, Fajtlowicz, 1988b, Fajtlowicz, 1990, Fajtlowicz, 1995, Fajtlowicz, 1998) and their fate, i.e., proof, refutation or ongoing open status. Graffiti builds conjectures of the form $i_{1} G ≤i_{2} G$ or $i_{1} G ≤i_{2} G +i_{3} G$ or $i_{1} G +i_{2} G ≤i_{3} G +i_{4} G$ , where the $i_{j} G$ are graph invariants, j=1,…,4, as well as, in later versions, forms involving ratios of invariants. These conjectures are then tested on a data-base of examples. Those which pass the test are numerous and sometimes of little interest. They are, therefore, submitted to a further series of procedures which eliminate presumably uninteresting ones because they follow from others by transitivity, or one invariant on the left-hand side is always smaller than one on the right-hand side, or they hold (i.e., are not refuted by the available examples) for larger classes of graphs than that one currently considered (e.g., bipartite graphs instead of trees) or they involve too close concepts, or they do not provide new information (for the current data-base).

Remaining conjectures are submitted to the mathematical community. They may be hard to prove or disprove. For instance of the eight open conjectures on the connectivity index described in the first paper on Graffiti (Fajtlowicz, 1988a) only two were recently disproved (with the help of the automated system AutoGraphiX, Caporossi and Hansen, 1997) and the six others remain open. Also in that paper are listed six theorems on the connectivity index, conjectured by Graffiti and proved by Chen or Fajtlowicz or both. Sometimes, proof of stronger results than those conjectured are obtained, e.g. that the connectivity index is not more than half the number of vertices, a result of Fajtlowicz (1997), while the system proposed this number as upper bound. Favaron et al. (1993) worked extensively on Graffiti’s conjectures and proved many results. Concerning the connectivity index they showed that for all graphs the index (or largest eigenvalue of the adjacency matrix) is at least as large as the ratio of the number of edges and the connectivity index. Moreover, the absolute value of any other eigenvalue of the adjacency matrix is never larger than the connectivity index (which implies three conjectures of Graffiti). Finally, for triangle-free graphs, the connectivity index is at least as large as the square root of the number of edges (Graffiti’s conjecture 213) which implies it is at least as large as the index (Graffiti’s conjecture 116). Shearer proved in 1988 (unpublished, Fajtlowicz, 1998, Shearer, 1999) that the average degree of a triangle-free graph is never larger than its connectivity index (Graffiti’s conjecture 63). Bollobás and Erdös (1998) proved the fundamental result that among all connected graphs G with a fixed number n of vertices the star has minimum connectivity index χ and therefore $χ G ≥ n−1 .$ They also proved that for all graphs G with a fixed number m of edges $χ G ≥ 14 8m+1 −1 .$ More than a dozen conjectures of Graffiti on the connectivity index remain open (Fajtlowicz, 1998). Otherwise, there are not too many published papers dealing with mathematical properties of χ (Maier, 1992, Araujo and Morales, 1996, Morales and Araujo, 1997).

Despite this activity, some fundamental questions related to the connectivity index are yet untouched, in particular concerning extremal graphs.

The aim of the present work is to contribute towards filling this gap. First of all, we show that among all n-vertex trees (and therefore also among all n-vertex acyclic molecular graphs) the path has the maximal χ-value. Bollobás and Erdös demonstrated that among all n-vertex trees the star has minimal χ-value. However, if n > 5 then the star is not a molecular graph. Hence the natural question: which acyclic molecular graphs have minimal connectivity indices? In the present paper, we address and resolve this problem.

In chemical graph theory (Trinajstić, 1992) acyclic molecular graphs are usually referred to as chemical trees. Thus, a chemical tree is a tree (= connected acyclic graph) in which no vertex has degree (= number of first neighbors) greater than four. Chemical trees are the graph representations of alkanes, or more precisely: of the carbon-atom skeletons of alkanes.

Let G be a molecular graph on n vertices. Let v be a vertex of G and let d_v be its degree. Recall that in a molecular graph (representing the carbon-atom skeleton of a saturated hydrocarbon), it is always true that 1≤d_v≤4.

The edge of G connecting the vertices u and v will be denoted by (uv). Then the connectivity index or Randić index of G is defined as $χ=χ G = ∑ uv 1 d_{u} d_{v}$ with summation embracing all edges of the graph G. Clearly, the connectivity-index-concept need not be restricted to molecular graphs, but its definition, Eq. (3), is applicable in the case of any graph.

Section snippets

Statement of the results

The main results of this work are summarized in the following two theorems.

The path P_n is the n-vertex tree with exactly two vertices of degree one. P_n is the molecular graph of the normal C_n-alkane, $CH_{3} CH_{2}_{n−2} CH_{3}$ . It is straightforward that $χ P_{n} = n−3 2 + 2 .$

Theorem 1

If T_n is any n-vertex chemical tree, n ≥ 2, then $χ(T_{n})≤χ(P_{n})$ with equality if and only if T_n is isomorphic to P_n.

As will be shown later, Theorem 1 remains valid if the word ‘chemical’ is deleted from its statement.

Theorem 2

Let T_n be any n-vertex chemical tree.

Search for chemical trees with extremal connectivity index

At the beginning of our research, we knew almost nothing about the structure of the minimum-χ chemical trees (whereas the form of the maximum-χ chemical tree was fairly obvious). In order to collect the necessary starting information we utilized our previously developed variable neighborhood search (VNS) metaheuristic (or general framework for building heuristics, Mladenović and Hansen, 1997, Hansen and Mladenović, 1997, Hansen and Mladenović, 1998a, Hansen and Mladenović, 1998b) which—in this

Chemical trees with minimal connectivity index are not unique

Consider an n-vertex chemical tree T_n and denote the number of its vertices of degree i by n_i, i = 1, 2, 3, 4. Further, denote the number of edges of T_n, connecting a vertex of degree i with a vertex of degree j by x_ij. Then, in accordance with Eq. (3), the connectivity index of T_n is given by $χ T_{n} = x_{12} 2 + x_{13} 3 + x_{14} 2 + x_{22} 2 + x_{23} 6 + x_{24} 22 + x_{33} 3 + x_{34} 23 + x_{44} 4$ From the expression (4) it is evident that whenever the values of the parameters x_ij are equal for two trees, then these trees have equal Randić index. In

Proof of theorem 1

In this section, we demonstrate that any n-vertex tree, different from the path P_n, has a connectivity index lower than $χ P_{n}$ . This, of course, implies the validity of Theorem 1.

In what follows, a vertex of degree one will be called a pendent vertex. An edge incident to a pendent vertex will be called a pendent edge.

Consider an n-vertex tree T_n, different from P_n. This tree must possess at least one branching vertex, that is a vertex of degree greater than two. Choose in T_n a branching vertex,

Proof of theorem 2

We demonstrate here the validity of Theorem 2 by employing a proof technique of linear programming (e.g., Chvátal, 1983). The notation used is the same as in Section 4. If T_n is an n-vertex tree, then the following six (linearly independent) relations are obeyed: $x_{12} +x_{13} +x_{14} =n_{1}$ $x_{12} +2x_{22} +x_{23} +x_{24} =2n_{2}$ $x_{13} +x_{23} +2x_{33} +x_{34} =3n_{3}$ $x_{14} +x_{24} +x_{34} +2x_{44} =4n_{4}$ $n_{1} +2n_{2} +3n_{3} +4n_{4} =2 n−1$ $n_{1} +n_{2} +n_{3} +n_{4} =n$ Our strategy is to consider , , , , , as a system of six linear equations in the unkowns n₁, n₂, n₃, n₄, x₁₄, x₄₄ and to solve

Acknowledgements

Work of the first and third authors supported by NSERC grant 105574, FCAR grant 32EQ1048 and a grant from CETAI-HEC. Work of the second author done in part during a visit to GERAD.

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Variable neighborhood search for extremal graphs: IV: Chemical trees with extremal connectivity index

Abstract

Introduction

Section snippets

Statement of the results

Search for chemical trees with extremal connectivity index

Chemical trees with minimal connectivity index are not unique

Proof of theorem 1

Proof of theorem 2

Acknowledgements

Chemical Physics Letters

Discrete Mathematics

Discrete Mathematics

Location Science

Computers and Operations Research

J. Mol. Struct. (Theorem)

Ars Combinatoria

Linear Programming

Journal of Chemical Information and Computer Sciences

Congressus Numerantium