Brief Communication: Graphs with maximum connectivity index

  • Authors:

  • Gilles Caporossi;Ivan Gutman;Pierre Hansen;Ljiljana Pavlović

  • Affiliations:

  • GERAD and ícole des Hautes ítudes Commerciales, 3000 Chemin de la Côte-Sainte-Catherine, Montréal, Canada H3T 2A7;Faculty of Science, University of Kragujevac, P.O. Box 60, YU-34000 Kragujevac, Yugoslavia;GERAD and ícole des Hautes ítudes Commerciales, 3000 Chemin de la Côte-Sainte-Catherine, Montréal, Canada H3T 2A7;Faculty of Science, University of Kragujevac, P.O. Box 60, YU-34000 Kragujevac, Yugoslavia

  • Venue:
  • Computational Biology and Chemistry
  • Year:
  • 2003

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Abstract

Let G be a graph and d"v the degree (=number of first neighbors) of its vertex v. The connectivity index of G is @g=@?(d"ud"v)^-^1^/^2, with the summation ranging over all pairs of adjacent vertices of G. In a previous paper (Comput. Chem. 23 (1999) 469), by applying a heuristic combinatorial optimization algorithm, the structure of chemical trees possessing extremal (maximum and minimum) values of @g was determined. It was demonstrated that the path P"n is the n-vertex tree with maximum @g-value. We now offer an alternative approach to finding (molecular) graphs with maximum @g, from which the extremality of P"n follows as a special case. By eliminating a flaw in the earlier proof, we demonstrate that there exist cases when @g does not provide a correct measure of branching: we find pairs of trees T,T', such that T is more branched than T', but @g(T)@g(T'). The smallest such examples are trees with 36 vertices in which one vertex has degree 31.