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On diameter and inverse degree of chemical graphs

Chen Xue-Gang (Department of Mathematics, North China Electric Power University, Beijing, China)
Fujita Shinya (Department of Integrated Design Engineering, Maebashi Institute of Technology, Kamisadori, Maebashi, Gunma, Japan)

The inverse degree r(G) of a finite graph G = (V,E) is defined as r(G) = Σv∈V 1/d(v), where d(v) is the degree of vertex v. In Discrete Math. 310 (2010), 940-946, Mukwembi posed the following conjecture: Let G be a connected chemical graph with diameter diam(G) and inverse degree r(G). Then diam(G) ≤ 12/5 r(G) + O(1). In this paper, we settle the conjecture affirmatively.

Keywords: inverse degree, diameter, chemical graph