scholarly journals Bounds on the first leap Zagreb index of trees

2021 ◽  
Vol 13 (2) ◽  
pp. 377-385
Author(s):  
N. Dehgardi ◽  
H. Aram
Keyword(s):  
Zagreb Index ◽  
Vertex Degrees

The first leap Zagreb index $LM1(G)$ of a graph $G$ is the sum of the squares of its second vertex degrees, that is, $LM_1(G)=\sum_{v\in V(G)}d_2(v/G)^2$, where $d_2(v/G)$ is the number of second neighbors of $v$ in $G$. In this paper, we obtain bounds for the first leap Zagreb index of trees and determine the extremal trees achieving these bounds.

scholarly journals Note on the Reformulated Zagreb Indices of Two Classes of Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-4
Author(s):  
Tongkun Qu ◽  
Mengya He ◽  
Shengjin Ji ◽  
Xia Li
Keyword(s):  
Upper Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Vertex Degrees

The reformulated Zagreb indices of a graph are obtained from the original Zagreb indices by replacing vertex degrees with edge degrees, where the degree of an edge is taken as the sum of degrees of its two end vertices minus 2. In this paper, we obtain two upper bounds of the first reformulated Zagreb index among all graphs with p pendant vertices and all graphs having key vertices for which they will become trees after deleting their one key vertex. Moreover, the corresponding extremal graphs which attained these bounds are characterized.

scholarly journals Molecular Descriptors of Nanotube, Oxide, Silicate, and Triangulene Networks

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Wei Gao ◽  
Muhammad Kamran Siddiqui
Keyword(s):  
Molecular Structure ◽  
Carbon Nanotube ◽  
Interconnection Networks ◽  
Chemical Structure ◽  
Chemical Reactivity ◽  
Hexagonal Lattice ◽  
Zagreb Index ◽  
Chemical Graph Theory ◽  
Vertex Degrees ◽  
Carbon Nanotube Networks

A topological index is a real number associated with chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity, or biological activity. The concept of hyper Zagreb index, first multiple Zagreb index, second multiple Zagreb index, and Zagreb polynomials was established in chemical graph theory based on vertex degrees. It is reported that these indices are useful in the study of anti-inflammatory activities of certain chemical networks. In this paper, we study carbon nanotube networks which are motivated by molecular structure of regular hexagonal lattice and also studied interconnection networks which are motivated by molecular structure of a chemical compound SiO4. We determine hyper Zagreb index, first multiple Zagreb index, second multiple Zagreb index, and Zagreb polynomials for some important class of carbon nanotube networks, dominating oxide network, dominating silicate network, and regular triangulene oxide network.

scholarly journals On topological properties of block shift and hierarchical hypercube networks

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 810-819
Author(s):  
Juan Luis García Guirao ◽  
Muhammad Kamran Siddiqui ◽  
Asif Hussain
Keyword(s):  
Graph Theory ◽  
Interconnection Networks ◽  
Communication Theory ◽  
Chemical Reactivity ◽  
Zagreb Index ◽  
Electronic Engineering ◽  
Chemical Graph Theory ◽  
Chemical Constitution ◽  
Vertex Degrees ◽  
Hypercube Networks

Abstract Networks play an important role in electrical and electronic engineering. It depends on what area of electrical and electronic engineering, for example there is a lot more abstract mathematics in communication theory and signal processing and networking etc. Networks involve nodes communicating with each other. Graph theory has found a considerable use in this area of research. A topological index is a real number associated with chemical constitution purporting for correlation of chemical networks with various physical properties, chemical reactivity. The concept of hyper Zagreb index, first multiple Zagreb index, second multiple Zagreb index and Zagreb polynomials was established in chemical graph theory based on vertex degrees. In this paper, we extend this study to interconnection networks and derive analytical closed results of hyper Zagreb index, first multiple Zagreb index, second multiple Zagreb index, Zagreb polynomials and redefined Zagreb indices for block shift network (BSN − 1) and (BSN − 2), hierarchical hypercube (HHC − 1) and (HHC − 2).

scholarly journals On the Zagreb index inequality of graphs with prescribed vertex degrees

2011 ◽  
Vol 159 (8) ◽  
pp. 852-858 ◽  
Author(s):  
Vesna Andova ◽  
Sašo Bogoev ◽  
Darko Dimitrov ◽  
Marcin Pilipczuk ◽  
Riste Škrekovski
Keyword(s):  
Zagreb Index ◽  
Vertex Degrees

scholarly journals On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs

2016 ◽  
Vol 14 (1) ◽  
pp. 641-648 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Spanning Trees ◽  
Line Graph ◽  
Upper And Lower Bounds ◽  
Barycentric Subdivision ◽  
Electrical Networks ◽  
Zagreb Index ◽  
Laplacian Eigenvalues ◽  
Estrada Index ◽  
Vertex Degrees ◽  
Number Of Spanning Trees

Abstract As a generalization of the Sierpiński-like graphs, the subdivided-line graph Г(G) of a simple connected graph G is defined to be the line graph of the barycentric subdivision of G. In this paper we obtain a closed-form formula for the enumeration of spanning trees in Г(G), employing the theory of electrical networks. We present bounds for the largest and second smallest Laplacian eigenvalues of Г(G) in terms of the maximum degree, the number of edges, and the first Zagreb index of G. In addition, we establish upper and lower bounds for the Laplacian Estrada index of Г(G) based on the vertex degrees of G. These bounds are also connected with the number of spanning trees in Г(G).

scholarly journals Relations between ordinary and multiplicative degree-based topological indices

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 3031-3042 ◽  
Author(s):  
Ivan Gutman ◽  
Igor Milovanovic ◽  
Emina Milovanovic
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Topological Indices ◽  
Upper And Lower Bounds ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Vertex Degrees ◽  
Simple Connected Graph

Let G be a simple connected graph with n vertices and m edges, and sequence of vertex degrees d1 ? d2 ?...? dn > 0. If vertices i and j are adjacent, we write i ~ j. Denote by ?1, ?*1, Q? and H? the multiplicative Zagreb index, multiplicative sum Zagreb index, general first Zagreb index, and general sumconnectivity index, respectively. These indices are defined as ?1 = ?ni=1 d2i, ?*1 = ?i~j(di+dj), Q? = ?n,i=1 d?i and H? = ?i~j(di+dj)?. We establish upper and lower bounds for the differences H?-m (?1*)?/m and Q?-n(?1)?/2n . In this way we generalize a number of results that were earlier reported in the literature.

scholarly journals F-Index of some graph operations

2016 ◽  
Vol 08 (02) ◽  
pp. 1650025 ◽  
Author(s):  
Nilanjan De ◽  
Sk. Md. Abu Nayeem ◽  
Anita Pal
Keyword(s):  
Electron Energy ◽  
Topological Index ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Molecular Graphs ◽  
Basic Properties ◽  
Graph Operations ◽  
Physico Chemical ◽  
Vertex Degrees

The F-index of a graph is defined as the sum of cubes of the vertex degrees of the graph. This was introduced in 1972, in the same paper where the first and second Zagreb indices were introduced to study the structure-dependency of total [Formula: see text]-electron energy. But this topological index was not further studied till then. Very recently, Furtula and Gutman [A forgotten topological index,J. Math. Chem. 53(4) (2015) 1184–1190.] reinvestigated the index and named it “forgotten topological index” or “F-index”. In that paper, they present some basic properties of this index and showed that this index can enhance the physico-chemical applicability of Zagreb index. Here, we study the behavior of this index under several graph operations and apply our results to find the F-index of different chemically interesting molecular graphs and nanostructures.

scholarly journals Graphs with maximal irregularity

Filomat ◽  
2014 ◽  
Vol 28 (7) ◽  
pp. 1315-1322 ◽  
Author(s):  
Hosam Abdo ◽  
Nathann Cohen ◽  
Darko Dimitrov
Keyword(s):  
Upper Bound ◽  
Undirected Graph ◽  
Graph Invariant ◽  
Zagreb Index ◽  
Second Zagreb Index ◽  
Chemical Graph Theory ◽  
Degree Of A Vertex ◽  
Maximal Vertex ◽  
Vertex Degrees ◽  
General Graphs

Albertson [3] has defined the P irregularity of a simple undirected graph G = (V,E) as irr(G) =?uv?E |dG(u)- dG(v)|, where dG(u) denotes the degree of a vertex u ? V. Recently, this graph invariant gained interest in the chemical graph theory, where it occured in some bounds on the first and the second Zagreb index, and was named the third Zagreb index [12]. For general graphs with n vertices, Albertson has obtained an asymptotically tight upper bound on the irregularity of 4n3/27: Here, by exploiting a different approach than in [3], we show that for general graphs with n vertices the upper bound ?n/3? ?2n/3? (?2n/3? -1) is sharp. We also present lower bounds on the maximal irregularity of graphs with fixed minimal and/or maximal vertex degrees, and consider an approximate computation of the irregularity of a graph.

scholarly journals Computing Topological Indices and Polynomials for Line Graphs

Mathematics ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 137 ◽  
Author(s):  
Shahid Imran ◽  
Muhammad Siddiqui ◽  
Muhammad Imran ◽  
Muhammad Nadeem
Keyword(s):  
Topological Index ◽  
Line Graph ◽  
Topological Indices ◽  
Line Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Second Zagreb Index ◽  
Vertex Degrees ◽  
Set Up ◽  
Ladder Graphs

A topological index is a number related to the atomic index that allows quantitative structure–action/property/toxicity connections. All the more vital topological indices correspond to certain physico-concoction properties like breaking point, solidness, strain vitality, and so forth, of synthetic mixes. The idea of the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials was set up in the substance diagram hypothesis in light of vertex degrees. These indices are valuable in the investigation of calming exercises of certain compound systems. In this paper, we computed the first and second Zagreb index, the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials of the line graph of wheel and ladder graphs by utilizing the idea of subdivision.

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.

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