On the Reformulated Multiplicative First Zagreb Index of Trees and Unicyclic Graphs

The multiplicative first Zagreb index of a graph H is defined as the product of the squares of the degrees of vertices of H . The line graph of a graph H is denoted by L H and is defined as the graph whose vertex set is the edge set of H where two vertices of L H are adjacent if and only if they are adjacent in H . The multiplicative first Zagreb index of the line graph of a graph H is referred to as the reformulated multiplicative first Zagreb index of H . This paper gives characterization of the unique graph attaining the minimum or maximum value of the reformulated multiplicative first Zagreb index in the class of all (i) trees of a fixed order (ii) connected unicyclic graphs of a fixed order.

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Efficient Algorithm for Generating Maximal L-Reflexive Trees

Symmetry ◽

10.3390/sym12050809 ◽

2020 ◽

Vol 12 (5) ◽

pp. 809

Author(s):

Milica Anđelić ◽

Dejan Živković

Keyword(s):

Efficient Algorithm ◽

Line Graph ◽

Common Vertex ◽

Line Graphs ◽

Algebraic Numbers ◽

Pisot Numbers ◽

Full Characterization ◽

Vertex Set ◽

Second Largest Eigenvalue

The line graph of a graph G is another graph of which the vertex set corresponds to the edge set of G, and two vertices of the line graph of G are adjacent if the corresponding edges in G share a common vertex. A graph is reflexive if the second-largest eigenvalue of its adjacency matrix is no greater than 2. Reflexive graphs give combinatorial ground to generate two classes of algebraic numbers, Salem and Pisot numbers. The difficult question of identifying those graphs whose line graphs are reflexive (called L-reflexive graphs) is naturally attacked by first answering this question for trees. Even then, however, an elegant full characterization of reflexive line graphs of trees has proved to be quite formidable. In this paper, we present an efficient algorithm for the exhaustive generation of maximal L-reflexive trees.

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On the first Zagreb index and multiplicative Zagreb coindices of graphs

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.1515/auom-2016-0008 ◽

2016 ◽

Vol 24 (1) ◽

pp. 153-176 ◽

Cited By ~ 3

Author(s):

Kinkar Ch. Das ◽

Nihat Akgunes ◽

Muge Togan ◽

Aysun Yurttas ◽

I. Naci Cangul ◽

...

Keyword(s):

Degree Sequence ◽

Molecular Graph ◽

Upper Bounds ◽

Extremal Graphs ◽

Lower And Upper Bounds ◽

First Zagreb Index ◽

Zagreb Index ◽

Irregularity Index ◽

Vertex Set ◽

The First Zagreb Index

AbstractFor a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as, where dG(vi) is the degree of vertex vi in G. Recently Xu et al. introduced two graphical invariantsandnamed as first multiplicative Zagreb coindex and second multiplicative Zagreb coindex, respectively. The Narumi-Katayama index of a graph G, denoted by NK(G), is equal to the product of the degrees of the vertices of G, that is, NK(G) =. The irregularity index t(G) of G is defined as the number of distinct terms in the degree sequence of G. In this paper, we give some lower and upper bounds on the first Zagreb index M1(G) of graphs and trees in terms of number of vertices, irregularity index, maxi- mum degree, and characterize the extremal graphs. Moreover, we obtain some lower and upper bounds on the (first and second) multiplicative Zagreb coindices of graphs and characterize the extremal graphs. Finally, we present some relations between first Zagreb index and Narumi-Katayama index, and (first and second) multiplicative Zagreb index and coindices of graphs.

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Topological indices of non-commuting graph of dihedral groups

Malaysian Journal of Fundamental and Applied Sciences ◽

10.11113/mjfas.v14n0.1270 ◽

2018 ◽

Vol 14 ◽

pp. 473-476 ◽

Cited By ~ 2

Author(s):

Nur Idayu Alimon ◽

Nor Haniza Sarmin ◽

Ahmad Erfanian

Keyword(s):

Wiener Index ◽

Topological Indices ◽

First Zagreb Index ◽

Dihedral Groups ◽

Zagreb Index ◽

Second Zagreb Index ◽

Commuting Graph ◽

Vertex Set ◽

Group A ◽

Group Of Symmetries

Assume is a non-abelian group A dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. The non-commuting graph of denoted by is the graph of vertex set whose vertices are non-central elements, in which is the center of and two distinct vertices and are joined by an edge if and only if In this paper, some topological indices of the non-commuting graph, of the dihedral groups, are presented. In order to determine the Edge-Wiener index, First Zagreb index and Second Zagreb index of the non-commuting graph, of the dihedral groups, previous results of some of the topological indices of non-commuting graph of finite group are used. Then, the non-commuting graphs of dihedral groups of different orders are found. Finally, the generalisation of Edge-Wiener index, First Zagreb index and Second Zagreb index of the non-commuting graphs of dihedral groups are determined.

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Logical Foundations of Kinematic Chains: Graphs, Line Graphs, and Hypergraphs

Journal of Mechanical Design ◽

10.1115/1.2912583 ◽

1990 ◽

Vol 112 (1) ◽

pp. 79-83 ◽

Cited By ~ 16

Author(s):

Frank Harary ◽

Hong-Sen Yan

Keyword(s):

Graph Theory ◽

Line Graph ◽

Kinematic Chain ◽

Line Graphs ◽

Kinematic Chains ◽

Logical Foundations ◽

Unique Graph ◽

Theory Of Graphs ◽

Admissible Graph

In terms of concepts from the theory of graphs and hypergraphs we formulate a precise structural characterization of a kinematic chain. To do this, we require the operations of line graph, intersection graph, and hypergraph duality. Using these we develop simple algorithms for constructing the unique graph G (KC) of a kinematic chain KC and (given an admissible graph G) for forming the unique kinematic chain whose graph is G. This one-to-one correspondence between kinematic chains and a class of graphs enables the mathematical and logical power, precision, concepts, and theorems of graph theory to be applied to gain new insights into the structure of kinematic chains.

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The first zagreb index and some hamiltonian properties of the line graph of a graph

Journal of Discrete Mathematical Sciences and Cryptography ◽

10.1080/09720529.2015.1103009 ◽

2017 ◽

Vol 20 (2) ◽

pp. 445-451 ◽

Cited By ~ 6

Author(s):

Rao Li ◽

Megan M. Taylor

Keyword(s):

Line Graph ◽

First Zagreb Index ◽

Zagreb Index ◽

Hamiltonian Properties ◽

The First Zagreb Index

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BALANCED UNITARY CAYLEY SIGRAPHS OVER FINITE COMMUTATIVE RINGS

Journal of Algebra and Its Applications ◽

10.1142/s0219498813501521 ◽

2014 ◽

Vol 13 (05) ◽

pp. 1350152 ◽

Cited By ~ 1

Author(s):

YOTSANAN MEEMARK ◽

BORWORN SUNTORNPOCH

Keyword(s):

Cayley Graph ◽

Line Graph ◽

Commutative Rings ◽

Signed Graph ◽

Group Of Units ◽

Vertex Set ◽

Finite Commutative Ring ◽

Unitary Cayley Graph ◽

Finite Commutative Rings

Let R be a finite commutative ring with identity 1. The unitary Cayley graph of R, denoted by GR, is the graph whose vertex set is R and the edge set {{a, b} : a, b ∈ R and a - b ∈ R×}, where R× is the group of units of R. We define the unitary Cayley signed graph (or unitary Cayley sigraph in short) to be an ordered pair 𝒮R = (GR, σ), where GR is the unitary Cayley graph over R with signature σ : E(GR) → {1, -1} given by [Formula: see text] In this paper, we give a criterion on R for SR to be balanced (every cycle in 𝒮R is positive) and a criterion for its line graph L(𝒮R) to be balanced. We characterize all finite commutative rings with the property that the marked sigraph 𝒮R,μ is canonically consistent. Moreover, we give a characterization of all finite commutative rings where 𝒮R, η(𝒮R) and L(𝒮R) are hyperenergetic balanced.

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Characterization of the logistic and loglogistic distributions by extreme value related stability with random sample size

Journal of Applied Probability ◽

10.1017/s0021900200116729 ◽

1987 ◽

Vol 24 (04) ◽

pp. 838-851 ◽

Cited By ~ 2

Author(s):

W. J. Voorn

Keyword(s):

Positive Integer ◽

Sample Size ◽

Random Variable ◽

Logistic Distribution ◽

Size Distributions ◽

Random Sample Size ◽

Maximum Value ◽

Maximum Stability ◽

Independent Observations

Maximum stability of a distribution with respect to a positive integer random variable N is defined by the property that the type of distribution is not changed when considering the maximum value of N independent observations. The logistic distribution is proved to be the only symmetric distribution which is maximum stable with respect to each member of a sequence of positive integer random variables assuming value 1 with probability tending to 1. If a distribution is maximum stable with respect to such a sequence and minimum stable with respect to another, then it must be logistic, loglogistic or ‘backward' loglogistic. The only possible sample size distributions in these cases are geometric.

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Topological characterization of dendrimer, benzenoid, and nanocone

Zeitschrift für Naturforschung C ◽

10.1515/znc-2018-0153 ◽

2018 ◽

Vol 74 (1-2) ◽

pp. 35-43

Author(s):

Wei Gao ◽

Muhammad Kamran Siddiqui ◽

Najma Abdul Rehman ◽

Mehwish Hussain Muhammad

Keyword(s):

Chemical Properties ◽

Chemical Compounds ◽

Topological Indices ◽

Zagreb Index ◽

Topological Characterization ◽

Complex Molecules ◽

Chemical Structures ◽

Physico Chemical ◽

Quantitative Structure Activity Relationships

Abstract Dendrimers are large and complex molecules with very well defined chemical structures. More importantly, dendrimers are highly branched organic macromolecules with successive layers or generations of branch units surrounding a central core. Topological indices are numbers associated with molecular graphs for the purpose of allowing quantitative structure-activity relationships. These topological indices correlate certain physico-chemical properties such as the boiling point, stability, strain energy, and others, of chemical compounds. In this article, we determine hyper-Zagreb index, first multiple Zagreb index, second multiple Zagreb index, and Zagreb polynomials for hetrofunctional dendrimers, triangular benzenoids, and nanocones.

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