scholarly journals Sharp bounds for the Zagreb indices of bicyclic graphs with k-pendant vertices

2010 ◽  
Vol 158 (17) ◽  
pp. 1953-1962 ◽  
Author(s):  
Qin Zhao ◽  
Shuchao Li
Keyword(s):  
Sharp Bounds ◽  
Zagreb Indices ◽  
Bicyclic Graphs

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

2017 ◽  
Vol 2017 ◽  
pp. 1-5 ◽  
Author(s):  
Wei Gao ◽  
Muhammad Kamran Jamil ◽  
Aisha Javed ◽  
Mohammad Reza Farahani ◽  
Shaohui Wang ◽  
...  
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Sharp Bounds ◽  
Zagreb Indices ◽  
Graph Transformations ◽  
Mathematical Properties ◽  
Bicyclic Graphs ◽  
Important Branch

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, whered(v)is the degree of the vertexvin a graphG=(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 amongn-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.

Sharp bounds for Zagreb indices of maximal outerplanar graphs

2010 ◽  
Vol 22 (2) ◽  
pp. 252-269 ◽  
Author(s):  
Ailin Hou ◽  
Shuchao Li ◽  
Lanzhen Song ◽  
Bing Wei
Keyword(s):  
Outerplanar Graphs ◽  
Sharp Bounds ◽  
Zagreb Indices ◽  
Maximal Outerplanar Graphs

Sharp bounds of the Zagreb indices of k-trees

2012 ◽  
Vol 27 (2) ◽  
pp. 271-291 ◽  
Author(s):  
John Estes ◽  
Bing Wei
Keyword(s):  
Sharp Bounds ◽  
Zagreb Indices

Extremal (n,m)-Graphs w.r.t General Multiplicative Zagreb Indices

2020 ◽  
Vol 23 ◽  
Author(s):  
Aisha Javed ◽  
Muhammad Kamran Jamil ◽  
Jia-Bao Liu ◽  
Akbar Ali
Keyword(s):  
Molecular Graph ◽  
Chemical Properties ◽  
Physical And Chemical Properties ◽  
Extremal Graphs ◽  
Unified Approach ◽  
Zagreb Indices ◽  
Graph Transformations ◽  
Bicyclic Graphs ◽  
Physical And Chemical ◽  
And Chemical Properties

Background:: A topological index of a molecular graph is the numeric quantity which can predict certain physical and chemical properties of the corresponding molecule. Xu et al. introduced some graph transformations which increase or decrease the first and second multiplicative Zagreb indices and proposed a unified approach to characterize extremal (n, m)- graphs. Method:: Graph transformations are used to find the extremal graphs, these transformations either increase or decrease the general multiplicative Zagreb indices. By applying the transformations which increase the general multiplicative Zagreb indices we find the graphs with maximal general multiplicative Zagreb indices and for minimal general Zagreb indices we use the transformations which decrease the index. Result:: In this paper, we extend the Xu’s results and show that the same graph transformations increase or decrease the first and second general multiplicative Zagreb indices for . As an application, the extremal acyclic, unicyclic and bicyclic graphs are presented for general multiplicative Zagreb indices. Conclusion:: By applying the transformation we investigated that in the class of acyclic, unicyclic and bicyclic graphs, which graph gives the minimum and the maximum general multiplicative Zagreb indices.

SOME RESULTS ON THE DIFFERENCE OF THE ZAGREB INDICES OF A GRAPH

2015 ◽  
Vol 92 (2) ◽  
pp. 177-186 ◽  
Author(s):  
MINGQIANG AN ◽  
LIMING XIONG
Keyword(s):  
Independence Number ◽  
Matching Number ◽  
Extremal Graphs ◽  
Vertex Connectivity ◽  
Zagreb Index ◽  
Sharp Bounds ◽  
Zagreb Indices ◽  
Second Zagreb Index ◽  
The Difference ◽  
Appl Math

The classical first and second Zagreb indices of a graph $G$ are defined as $M_{1}(G)=\sum _{v\in V(G)}d(v)^{2}$ and $M_{2}(G)=\sum _{e=uv\in E(G)}d(u)d(v),$ where $d(v)$ is the degree of the vertex $v$ of $G.$ Recently, Furtula et al. [‘On difference of Zagreb indices’, Discrete Appl. Math.178 (2014), 83–88] studied the difference of $M_{1}$ and $M_{2},$ and showed that this difference is closely related to the vertex-degree-based invariant $RM_{2}(G)=\sum _{e=uv\in E(G)}[d(u)-1][d(v)-1]$, the reduced second Zagreb index. In this paper, we present sharp bounds for the reduced second Zagreb index, given the matching number, independence number and vertex connectivity, and we also completely determine the extremal graphs.

scholarly journals Sharp bounds on Zagreb indices of cacti with k pendant vertices

Filomat ◽  
2012 ◽  
Vol 26 (6) ◽  
pp. 1189-1200 ◽  
Author(s):  
Shuchao Li ◽  
Huangxu Yang ◽  
Qin Zhao
Keyword(s):  
Perfect Matching ◽  
Molecular Graph ◽  
Common Vertex ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Sharp Bounds ◽  
Zagreb Indices ◽  
Sum Of Products ◽  
Vertex Degrees ◽  
The First Zagreb Index

For a (molecular) graph, the first Zagreb index M1 is equal to the sum of squares of its vertex degrees, and the second Zagreb index M2 is equal to the sum of products of degrees of pairs of adjacent vertices. A connected graph G is a cactus if any two of its cycles have at most one common vertex. In this paper, we investigate the first and the second Zagreb indices of cacti with k pendant vertices. We determine sharp bounds for M1 -, M2 -values of n-vertex cacti with k pendant vertices. As a consequence, we determine the n-vertex cacti with maximal Zagreb indices and we also determine the cactus with a perfect matching having maximal Zagreb indices.

scholarly journals Zagreb Indices of Trees, Unicyclic and Bicyclic Graphs With Given (Total) Domination

IEEE Access ◽  
2019 ◽  
Vol 7 ◽  
pp. 94143-94149 ◽  
Author(s):  
Doost Ali Mojdeh ◽  
Mohammad Habibi ◽  
Leila Badakhshian ◽  
Yongsheng Rao
Keyword(s):  
Total Domination ◽  
Zagreb Indices ◽  
Bicyclic Graphs
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