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

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.

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Multiplicative Zagreb indices of cacti

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830916500403 ◽

2016 ◽

Vol 08 (03) ◽

pp. 1650040 ◽

Cited By ~ 6

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.

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

Discrete Dynamics in Nature and Society ◽

10.1155/2017/6079450 ◽

2017 ◽

Vol 2017 ◽

pp. 1-5 ◽

Cited By ~ 6

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.

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Relations between ordinary and multiplicative degree-based topological indices

Filomat ◽

10.2298/fil1808031g ◽

2018 ◽

Vol 32 (8) ◽

pp. 3031-3042 ◽

Cited By ~ 1

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.

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Tetracyclic graphs with maximum second Zagreb index: A simple approach

Asian-European Journal of Mathematics ◽

10.1142/s179355711850064x ◽

2018 ◽

Vol 11 (05) ◽

pp. 1850064 ◽

Cited By ~ 5

Author(s):

Akbar Ali

Keyword(s):

Graph Theory ◽

Short Note ◽

Topological Indices ◽

Simple Approach ◽

Graph Invariants ◽

Zagreb Index ◽

Chemical Graph ◽

Zagreb Indices ◽

Second Zagreb Index ◽

Chemical Graph Theory

In the chemical graph theory, graph invariants are usually referred to as topological indices. The second Zagreb index (denoted by [Formula: see text]) is one of the most studied topological indices. For [Formula: see text], let [Formula: see text] be the collection of all non-isomorphic connected graphs with [Formula: see text] vertices and [Formula: see text] edges (such graphs are known as tetracyclic graphs). Recently, Habibi et al. [Extremal tetracyclic graphs with respect to the first and second Zagreb indices, Trans. on Combin. 5(4) (2016) 35–55.] characterized the graph having maximum [Formula: see text] value among all members of the collection [Formula: see text]. In this short note, an alternative but relatively simple approach is used for characterizing the aforementioned graph.

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Lower and Upper Bounds for Hyper-Zagreb Index of Graphs

Iraqi Journal of Science ◽

10.24996/ijs.2020.61.6.18 ◽

2020 ◽

pp. 1401-1406

Author(s):

G. H. SHIRDEL ◽

H. REZAPOUR ◽

R. NASIRI

Keyword(s):

Lower Bounds ◽

Connected Graph ◽

Chemical Compounds ◽

Upper Bounds ◽

Topological Indices ◽

Upper And Lower Bounds ◽

Lower And Upper Bounds ◽

Zagreb Index ◽

Simple Connected Graph

The topological indices are functions on the graph that do not depend on the labeling of their vertices. They are used by chemists for studying the properties of chemical compounds. Let be a simple connected graph. The Hyper-Zagreb index of the graph , is defined as ,where and are the degrees of vertex and , respectively. In this paper, we study the Hyper-Zagreb index and give upper and lower bounds for .

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Sharp Upper and Lower Bounds of VDB Topological Indices of Digraphs

Symmetry ◽

10.3390/sym13101903 ◽

2021 ◽

Vol 13 (10) ◽

pp. 1903

Author(s):

Juan Monsalve ◽

Juan Rada

Keyword(s):

Lower Bounds ◽

Topological Index ◽

General Setting ◽

Upper Bounds ◽

Topological Indices ◽

Vertex Degree ◽

Upper And Lower Bounds ◽

Lower And Upper Bounds ◽

Extremal Values

A vertex-degree-based (VDB, for short) topological index φ induced by the numbers φij was recently defined for a digraph D, as φD=12∑uvφdu+dv−, where du+ denotes the out-degree of the vertex u,dv− denotes the in-degree of the vertex v, and the sum runs over the set of arcs uv of D. This definition generalizes the concept of a VDB topological index of a graph. In a general setting, we find sharp lower and upper bounds of a symmetric VDB topological index over Dn, the set of all digraphs with n non-isolated vertices. Applications to well-known topological indices are deduced. We also determine extremal values of symmetric VDB topological indices over OTn and OG, the set of oriented trees with n vertices, and the set of all orientations of a fixed graph G, respectively.

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Bounds on the weighted vertex PI index of cacti graphs

Filomat ◽

10.2298/fil1918977m ◽

2019 ◽

Vol 33 (18) ◽

pp. 5977-5989

Author(s):

Gang Ma ◽

Qiuju Bian ◽

Jianfeng Wang

Keyword(s):

Lower Bounds ◽

Vertex Degree ◽

Upper And Lower Bounds ◽

Extremal Graphs ◽

Pi Index ◽

Weighted Vertex

The weighted vertex PI index of a graph G is defined by PIw(G)=?e=uv?E(G) (dG(u)+dG(v))(nu(e|G) + nv(e|G)) where dG(u) denotes the vertex degree of u and nu(e|G) denotes the number of vertices in G whose distance to the vertex u is smaller than the distance to the vertex v. A graph is a cactus if it is connected and all its blocks are either edges or cycles. In this paper, we give the upper and lower bounds on the weighted vertex PI index of cacti with n vertices and s cycles, and completely characterize the corresponding extremal graphs.

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On the Generalized Distance Energy of Graphs

Mathematics ◽

10.3390/math8010017 ◽

2019 ◽

Vol 8 (1) ◽

pp. 17 ◽

Cited By ~ 4

Author(s):

Abdollah Alhevaz ◽

Maryam Baghipur ◽

Hilal A. Ganie ◽

Yilun Shang

Keyword(s):

Lower Bounds ◽

Complete Graph ◽

Connected Graph ◽

Distance Matrix ◽

Wiener Index ◽

Diagonal Matrix ◽

Upper And Lower Bounds ◽

Star Graph ◽

Extremal Graphs ◽

Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

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Wiener–Hosoya Matrix of Connected Graphs

Mathematics ◽

10.3390/math9040359 ◽

2021 ◽

Vol 9 (4) ◽

pp. 359

Author(s):

Hassan Ibrahim ◽

Reza Sharafdini ◽

Tamás Réti ◽

Abolape Akwu

Keyword(s):

Lower Bounds ◽

Molecular Graph ◽

Wiener Index ◽

Vertex Degree ◽

Upper And Lower Bounds ◽

Connected Graphs ◽

Matrix Description ◽

Vertex Set

Let G be a connected (molecular) graph with the vertex set V(G)={v1,⋯,vn}, and let di and σi denote, respectively, the vertex degree and the transmission of vi, for 1≤i≤n. In this paper, we aim to provide a new matrix description of the celebrated Wiener index. In fact, we introduce the Wiener–Hosoya matrix of G, which is defined as the n×n matrix whose (i,j)-entry is equal to σi2di+σj2dj if vi and vj are adjacent and 0 otherwise. Some properties, including upper and lower bounds for the eigenvalues of the Wiener–Hosoya matrix are obtained and the extremal cases are described. Further, we introduce the energy of this matrix.

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Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders

Open Mathematics ◽

10.1515/math-2019-0053 ◽

2019 ◽

Vol 17 (1) ◽

pp. 668-676

Author(s):

Tingzeng Wu ◽

Huazhong Lü

Keyword(s):

Lower Bounds ◽

Connected Graph ◽

Wiener Index ◽

Upper And Lower Bounds ◽

Extremal Graphs ◽

Recurrence Formulae

Abstract Let G be a connected graph and u and v two vertices of G. The hyper-Wiener index of graph G is $\begin{array}{} WW(G)=\frac{1}{2}\sum\limits_{u,v\in V(G)}(d_{G}(u,v)+d^{2}_{G}(u,v)) \end{array}$, where dG(u, v) is the distance between u and v. In this paper, we first give the recurrence formulae for computing the hyper-Wiener indices of polyphenyl chains and polyphenyl spiders. We then obtain the sharp upper and lower bounds for the hyper-Wiener index among polyphenyl chains and polyphenyl spiders, respectively. Moreover, the corresponding extremal graphs are determined.

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