The Harary index of ordinary and generalized quasi-tree graphs

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The quasi-tree graph is a graph G in which there exists a vertex v ∈ V ( G ) such that G − v is a tree. In this paper, we presented the upper and lower bounds on the Harary index of...

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Published in	Journal of applied mathematics & computing Vol. 45; no. 1-2; pp. 365 - 374
Main Authors	Xu, Kexiang, Wang, Jinlan, Liu, Hongshuang
Format	Journal Article
Language	English
Published	Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2014 Springer Nature B.V
Subjects	Applied mathematics Computation Computational Mathematics and Numerical Analysis Equality Graph theory Graphs Lower bounds Mathematical analysis Mathematical and Computational Engineering Mathematical models Mathematics Mathematics and Statistics Mathematics of Computing Original Research Studies Theory of Computation Trees 05C90 05C12 Quasi-tree graph Distance (in graph) 05C05 Harary index
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Abstract	The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The quasi-tree graph is a graph G in which there exists a vertex v ∈ V ( G ) such that G − v is a tree. In this paper, we presented the upper and lower bounds on the Harary index of all quasi-tree graphs of order n and characterized the corresponding extremal graphs. Moreover we defined the k -generalized quasi-tree graph to be a connected graph G with a subset V k ⊆ V ( G ) where \| V k \|= k such that G − V k is a tree. And we also determined the k -generalized quasi-tree graph of order n with maximal Harary index for all values of k and the extremal one with minimal Harary index for k =2.
AbstractList	The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The quasi-tree graph is a graph G in which there exists a vertex vV(G) such that G-v is a tree. In this paper, we presented the upper and lower bounds on the Harary index of all quasi-tree graphs of order n and characterized the corresponding extremal graphs. Moreover we defined the k-generalized quasi-tree graph to be a connected graph G with a subset V ^sub k^V(G) where \|V ^sub k^\|=k such that G-V ^sub k^ is a tree. And we also determined the k-generalized quasi-tree graph of order n with maximal Harary index for all values of k and the extremal one with minimal Harary index for k=2.[PUBLICATION ABSTRACT] The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The quasi-tree graph is a graph G in which there exists a vertex vV(G) such that G-v is a tree. In this paper, we presented the upper and lower bounds on the Harary index of all quasi-tree graphs of order n and characterized the corresponding extremal graphs. Moreover we defined the k-generalized quasi-tree graph to be a connected graph G with a subset V sub( )kiVG) where \|V sub( )k=k such that G-V sub( )kis a tree. And we also determined the k-generalized quasi-tree graph of order n with maximal Harary index for all values of k and the extremal one with minimal Harary index for k=2. The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The quasi-tree graph is a graph G in which there exists a vertex v ∈ V ( G ) such that G − v is a tree. In this paper, we presented the upper and lower bounds on the Harary index of all quasi-tree graphs of order n and characterized the corresponding extremal graphs. Moreover we defined the k -generalized quasi-tree graph to be a connected graph G with a subset V k ⊆ V ( G ) where \| V k \|= k such that G − V k is a tree. And we also determined the k -generalized quasi-tree graph of order n with maximal Harary index for all values of k and the extremal one with minimal Harary index for k =2.
Author	Xu, Kexiang Liu, Hongshuang Wang, Jinlan
Author_xml	– sequence: 1 givenname: Kexiang surname: Xu fullname: Xu, Kexiang email: kexxu1221@126.com organization: College of Science, Nanjing University of Aeronautics & Astronautics – sequence: 2 givenname: Jinlan surname: Wang fullname: Wang, Jinlan organization: College of Science, Nanjing University of Aeronautics & Astronautics – sequence: 3 givenname: Hongshuang surname: Liu fullname: Liu, Hongshuang organization: College of Science, Nanjing University of Aeronautics & Astronautics
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Cites_doi	10.1023/A:1016290123303 10.1007/BF01164642 10.1007/s10910-007-9339-2 10.1016/j.dam.2011.06.003 10.1023/A:1010767517079 10.1021/ja01193a005 10.1007/978-1-349-03521-2 10.1186/1029-242X-2013-1 10.1007/BF01164638 10.1016/j.dam.2011.08.014 10.1016/0009-2614(93)87094-J 10.1002/9783527628766 10.1021/ci00023a007 10.1007/s10910-009-9520-x 10.1002/9783527613106 10.1016/j.aml.2010.04.017
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Snippet	The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The quasi-tree graph is a graph G in... The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The quasi-tree graph is a graph G in...
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SubjectTerms	Applied mathematics Computation Computational Mathematics and Numerical Analysis Equality Graph theory Graphs Lower bounds Mathematical analysis Mathematical and Computational Engineering Mathematical models Mathematics Mathematics and Statistics Mathematics of Computing Original Research Studies Theory of Computation Trees
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Title	The Harary index of ordinary and generalized quasi-tree graphs
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