scholarly journals On the sharp lower bounds of Zagreb indices of graphs with given number of cut vertices

2018 ◽  
Vol 458 (1) ◽  
pp. 21-29 ◽  
Author(s):  
Shengjin Ji ◽  
Shaohui Wang
Keyword(s):  
Lower Bounds ◽  
Zagreb Indices

scholarly journals Multiplicative Zagreb indices of cacti

2016 ◽  
Vol 08 (03) ◽  
pp. 1650040 ◽  
Author(s):  
Shaohui Wang ◽  
Bing Wei
Keyword(s):  
Graph Theory ◽  
Lower Bounds ◽  
Connected Graph ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Material Chemistry ◽  
Cactus Graph ◽  
Cactus Graphs

Let [Formula: see text] be multiplicative Zagreb index of a graph [Formula: see text]. A connected graph is a cactus graph if and only if any two of its cycles have at most one vertex in common, which is a generalization of trees and has been the interest of researchers in the field of material chemistry and graph theory. In this paper, we use a new tool to obtain the upper and lower bounds of [Formula: see text] for all cactus graphs and characterize the corresponding extremal 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.

scholarly journals Some properties of the Zagreb indices

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2667-2675
Author(s):  
Emina Milovanovic ◽  
Igor Milovanovic ◽  
Muhammad Jamil
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Simple Graph ◽  
Graph Invariants ◽  
Zagreb Indices

Let G = (V,E), V = {1,2,..., n}, E = {e1,e2,..., em}, be a simple graph with n vertices and m edges. Denote by d1 ? d2 ? ... ? dn > 0, and d(e1) ? d(e2) ? d(em), sequences of vertex and edge degrees, respectively. If i-th and j-th vertices of G are adjacent, it is denoted as i ~ j. Graph invariants referred to as the first, second and the first reformulated Zagreb indices are defined as M1=?ni=1 di2, M2= ?i~j didj and EM1 = ?mi=1 d(ei)2, respectively. Let ?1 ? ?2? ... ?n be eigenvalues of G. With ?(G) = ?1 a spectral radius of G is denoted. Lower bounds for invariants M1, M2, EM1 and ?(G) are obtained.

scholarly journals Sharp lower bounds for the Zagreb indices of unicyclic graphs

2015 ◽  
Vol 39 ◽  
pp. 595-603 ◽  
Author(s):  
Batmend HOROLDAGVA ◽  
Kinkar Ch. DAS
Keyword(s):  
Lower Bounds ◽  
Unicyclic Graphs ◽  
Zagreb Indices

scholarly journals Lower Bounds on the Entire Zagreb Indices of Trees

2020 ◽  
Vol 2020 ◽  
pp. 1-8 ◽  
Author(s):  
Liang Luo ◽  
Nasrin Dehgardi ◽  
Asfand Fahad
Keyword(s):  
Lower Bounds ◽  
Molecular Graph ◽  
Zagreb Indices ◽  
Degree Of A Vertex

For a (molecular) graph G, the first and the second entire Zagreb indices are defined by the formulas M1εG=∑x∈VG∪EGdx2 and M2εG=∑x is either adjacent or incident to ydxdy in which dx represents the degree of a vertex or an edge x. In the current manuscript, we establish some lower bounds on the first and the second entire Zagreb indices and determine the extremal trees which achieve these bounds.

scholarly journals Bounds and extremal graphs of second reformulated Zagreb index for graphs with cyclomatic number at most three

2021 ◽  
Vol 49 (1) ◽  
Author(s):  
Abhay Rajpoot ◽  
◽  
Lavanya Selvaganesh ◽  
Keyword(s):  
Lower Bounds ◽  
Topological Indices ◽  
Simple Approach ◽  
Vertex Degree ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Cyclomatic Number ◽  
Tricyclic Graphs

Miliˇcevi´c et al., in 2004, introduced topological indices known as Reformulated Zagreb indices, where they modified Zagreb indices using the edge-degree instead of vertex degree. In this paper, we present a simple approach to find the upper and lower bounds of the second reformulated Zagreb index, EM2(G), by using six graph operations/transformations. We prove that these operations significantly alter the value of reformulated Zagreb index. We apply these transformations and identify those graphs with cyclomatic number at most 3, namely trees, unicyclic, bicyclic and tricyclic graphs, which attain the upper and lower bounds of second reformulated Zagreb index for graphs.

Two Lower Bounds for Shortest Double-Base Number System

2015 ◽  
Vol E98.A (6) ◽  
pp. 1310-1312 ◽  
Author(s):  
Parinya CHALERMSOOK ◽  
Hiroshi IMAI ◽  
Vorapong SUPPAKITPAISARN
Keyword(s):  
Lower Bounds ◽  
Number System ◽  
Base Number ◽  
Double Base
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