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Volume 43 Number 5 Volume 43 Number 6

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Research articles

ScienceAsia 43 (2017): 387-393 |doi: 10.2306/scienceasia1513-1874.2017.43.387

The general zeroth-order Randić index of maximal outerplanar graphs and trees with k maximum degree vertices

Guifu Sua, Minghui Menga, Lihong Cuia,*, Zhibing Chenb, Lan Xuc

ABSTRACT:     For a graph, the general zeroth-order Randić index R0α is defined as the sum of the αth power of the vertex degrees (α≠0, α≠1). Let ℋn be the class of all maximal outerplanar graphs on n vertices, and Tn,k be the class of trees with n vertices of which k vertices have the maximum degree. We first present a lower bound (respectively, upper bound) for the general zeroth-order Randić index of graphs in ℋn (respectively, Tn,k) when α∈(−∞,0)∪(1,+∞) (respectively, α∈(2,+∞)), and characterize the extremal graphs. Then we determine graphs of the class Tn,k with maximal and minimal general zeroth-order Randić index when α∈(−∞,0)∪(1,+∞), respectively.

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a School of Science, Beijing University of Chemical Technology, Beijing 100029, China
b College of Mathematics and Statistics, Shenzhen University, Guangdong 518060, China
c Department of Mathematics, Changji University, Changji 831100, China

* Corresponding author, E-mail: gfs@mail.buct.edu.cn

Received 17 Jul 2017, Accepted 24 Oct 2017