Sharp Bounds of the Hyper-Zagreb Index on Acyclic, Unicylic, and Bicyclic Graphs

Sharp Bounds of the Hyper-Zagreb Index on Acyclic, Unicylic, and Bicyclic Graphs

The hyper-Zagreb index is an important branch in the Zagreb indices family, which is defined as ∑uv∈E(G)‍(d(u)+d(v))2, where d(v) is the degree of the vertex v in a graph G=(V(G),E(G)). In this paper, the monotonicity of the hyper-Zagreb index under some graph transformations was studied. Using thes...

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Main Authors: Wei Gao, Muhammad Kamran Jamil, Aisha Javed, Mohammad Reza Farahani, Shaohui Wang, Jia-Bao Liu
Format: Article
Language:English
Published: Hindawi Limited 2017-01-01
Series:Discrete Dynamics in Nature and Society
Online Access:http://dx.doi.org/10.1155/2017/6079450
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spelling doaj-8a14e942171342528621223e3fcdda012020-11-24T21:02:56ZengHindawi LimitedDiscrete Dynamics in Nature and Society1026-02261607-887X2017-01-01201710.1155/2017/60794506079450Sharp Bounds of the Hyper-Zagreb Index on Acyclic, Unicylic, and Bicyclic GraphsWei Gao0Muhammad Kamran Jamil1Aisha Javed2Mohammad Reza Farahani3Shaohui Wang4Jia-Bao Liu5School of Information Science and Technology, Yunnan Normal University, Kunming 650500, ChinaRiphah Institute of Computing and Applied Sciences (RICAS), Riphah International University, Lahore, PakistanAbdus Salam School of Mathematical Sciences, Government College University, Lahore, PakistanDepartment of Applied Mathematics, Iran University of Science and Technology, Narmak, Tehran, IranDepartment of Mathematics and Computer Science, Adelphi University, Garden City, NY 11530, USASchool of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, ChinaThe hyper-Zagreb index is an important branch in the Zagreb indices family, which is defined as ∑uv∈E(G)‍(d(u)+d(v))2, where d(v) is the degree of the vertex v in a graph G=(V(G),E(G)). In this paper, the monotonicity of the hyper-Zagreb index under some graph transformations was studied. Using these nice mathematical properties, the extremal graphs among n-vertex trees (acyclic), unicyclic, and bicyclic graphs are determined for hyper-Zagreb index. Furthermore, the sharp upper and lower bounds on the hyper-Zagreb index of these graphs are provided.http://dx.doi.org/10.1155/2017/6079450
collection DOAJ
language English
format Article
sources DOAJ
author Wei Gao
Muhammad Kamran Jamil
Aisha Javed
Mohammad Reza Farahani
Shaohui Wang
Jia-Bao Liu
spellingShingle Wei Gao
Muhammad Kamran Jamil
Aisha Javed
Mohammad Reza Farahani
Shaohui Wang
Jia-Bao Liu
Sharp Bounds of the Hyper-Zagreb Index on Acyclic, Unicylic, and Bicyclic Graphs
Discrete Dynamics in Nature and Society
author_facet Wei Gao
Muhammad Kamran Jamil
Aisha Javed
Mohammad Reza Farahani
Shaohui Wang
Jia-Bao Liu
author_sort Wei Gao
title Sharp Bounds of the Hyper-Zagreb Index on Acyclic, Unicylic, and Bicyclic Graphs
title_short Sharp Bounds of the Hyper-Zagreb Index on Acyclic, Unicylic, and Bicyclic Graphs
title_full Sharp Bounds of the Hyper-Zagreb Index on Acyclic, Unicylic, and Bicyclic Graphs
title_fullStr Sharp Bounds of the Hyper-Zagreb Index on Acyclic, Unicylic, and Bicyclic Graphs
title_full_unstemmed Sharp Bounds of the Hyper-Zagreb Index on Acyclic, Unicylic, and Bicyclic Graphs
title_sort sharp bounds of the hyper-zagreb index on acyclic, unicylic, and bicyclic graphs
publisher Hindawi Limited
series Discrete Dynamics in Nature and Society
issn 1026-0226
1607-887X
publishDate 2017-01-01
description The hyper-Zagreb index is an important branch in the Zagreb indices family, which is defined as ∑uv∈E(G)‍(d(u)+d(v))2, where d(v) is the degree of the vertex v in a graph G=(V(G),E(G)). In this paper, the monotonicity of the hyper-Zagreb index under some graph transformations was studied. Using these nice mathematical properties, the extremal graphs among n-vertex trees (acyclic), unicyclic, and bicyclic graphs are determined for hyper-Zagreb index. Furthermore, the sharp upper and lower bounds on the hyper-Zagreb index of these graphs are provided.
url http://dx.doi.org/10.1155/2017/6079450
work_keys_str_mv AT weigao sharpboundsofthehyperzagrebindexonacyclicunicylicandbicyclicgraphs
AT muhammadkamranjamil sharpboundsofthehyperzagrebindexonacyclicunicylicandbicyclicgraphs
AT aishajaved sharpboundsofthehyperzagrebindexonacyclicunicylicandbicyclicgraphs
AT mohammadrezafarahani sharpboundsofthehyperzagrebindexonacyclicunicylicandbicyclicgraphs
AT shaohuiwang sharpboundsofthehyperzagrebindexonacyclicunicylicandbicyclicgraphs
AT jiabaoliu sharpboundsofthehyperzagrebindexonacyclicunicylicandbicyclicgraphs
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