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Erschienen in:
01.06.2012 | Discrete and combinatorial mathematic
The starlike trees with maximum and minimum second order connectivity index
Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2012
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Abstract
Let G be a simple graph and h≥0 be an integer. The higher order connectivity index
R
h
(G) of G is defined as
where d
i
denotes the degree of the vertex v
i
and \(v_{i_{1}}v_{i_{2}}\cdots v_{i_{h+1}}\) runs over all paths of length h in G. A starlike tree is a tree with unique vertex of degree greater than two. Rada and Araujo proved that the starlike trees which have equal connectivity index of order h for all h≥0 are isomorphic. By T(n) we denote the set of the starlike trees on n vertices. In this paper, we characterize the extremal starlike trees with maximum and minimum second order connectivity index in T(n).
$$R_h(G)=\sum_{v_{i_1}v_{i_2}\cdots v_{i_{h+1}}} \frac{1}{\sqrt {d_{i_1}d_{i_2}\cdots d_{i_{h+1}}}},$$