scholarly journals The sharp bounds of Zagreb indices on connected graphs

2021 ◽  
pp. 477-489
Author(s):  
Yuming Chu ◽  
Muhammad Kashif Shafiq ◽  
Muhammad Imran ◽  
Muhammad Kamran Siddiqui ◽  
Hafiz Muhammad Afzal Siddiqui ◽  
...  
Keyword(s):  
Sharp Bounds ◽  
Connected Graphs ◽  
Zagreb Indices

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

The local partition dimension of graphs

2020 ◽  
pp. 2150028 ◽  
Author(s):  
Ridho Alfarisi ◽  
Arika Indah Kristiana ◽  
Dafik
Keyword(s):  
Sharp Bounds ◽  
Connected Graphs ◽  
Partition Dimension

All graphs in this paper are undirected and connected graphs. An ordered k-partition set [Formula: see text] where [Formula: see text], the representation of a vertex [Formula: see text] of [Formula: see text] with respect to [Formula: see text] is [Formula: see text] where [Formula: see text] is the distance between the vertex v and the set [Formula: see text] with [Formula: see text] for [Formula: see text]. The partition set [Formula: see text] is a local resolving partition of [Formula: see text] if [Formula: see text] for [Formula: see text] adjacent to [Formula: see text] of [Formula: see text]. The minimum local resolving partition [Formula: see text] is a local partition dimension of [Formula: see text], denoted by [Formula: see text]. In our paper, we found the sharp bounds of the local partition dimension of [Formula: see text] and determine the exact value of some special graph.

Product version of reciprocal Gutman indices of composite graphs

2017 ◽  
Vol 26 (2) ◽  
pp. 211-219
Author(s):  
K. Pattabiraman
Keyword(s):  
Tensor Product ◽  
Upper Bounds ◽  
Graph Invariants ◽  
Harary Index ◽  
Connected Graphs ◽  
Zagreb Indices ◽  
Strong Product

In this paper, we present the upper bounds for the product version of reciprocal Gutman indices of the tensor product, join and strong product of two connected graphs in terms of other graph invariants including the Harary index and Zagreb indices.

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 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

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 Comparing Zagreb indices for connected graphs

2010 ◽  
Vol 158 (10) ◽  
pp. 1073-1078 ◽  
Author(s):  
Batmend Horoldagva ◽  
Sang-Gu Lee
Keyword(s):  
Connected Graphs ◽  
Zagreb Indices
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