VERTEX WEIGHTED WIENER POLYNOMIALS OF THE SUBDIVISION GRAPH AND THE LINE GRAPH OF SUBDIVISION GRAPH OF THE WHEEL GRAPH

Given a connected regular graph $G$, let $l(G)$ be its line graph, $s(G)$ its subdivision graph, $r(G)$ the graph obtained from $G$ by adding a new vertex corresponding to each edge of $G$ and joining each new vertex to the end vertices of the corresponding edge and $q(G)$ the graph obtained from $G$ by inserting a new vertex into every edge of $G$ and new edges joining the pairs of new vertices which lie on adjacent edges of $G$. A formula for the normalised Laplacian characteristic polynomial of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$) in terms of the normalised Laplacian characteristic polynomial of $G$ and the number of vertices and edges of $G$ is developed and used to give a sharp lower bound for the degree-Kirchhoff index and a formula for the number of spanning trees of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$).

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Hyperbolicity on Graph Operators

Symmetry ◽

10.3390/sym10090360 ◽

2018 ◽

Vol 10 (9) ◽

pp. 360 ◽

Cited By ~ 2

Author(s):

J. Méndez-Bermúdez ◽

Rosalío Reyes ◽

José Rodríguez ◽

José Sigarreta

Keyword(s):

Line Graph ◽

Discrete Mathematics ◽

Total Graph ◽

Subdivision Graph ◽

Gromov Product

A graph operator is a mapping F : Γ → Γ ′ , where Γ and Γ ′ are families of graphs. The different kinds of graph operators are an important topic in Discrete Mathematics and its applications. The symmetry of this operations allows us to prove inequalities relating the hyperbolicity constants of a graph G and its graph operators: line graph, Λ ( G ) ; subdivision graph, S ( G ) ; total graph, T ( G ) ; and the operators R ( G ) and Q ( G ) . In particular, we get relationships such as δ ( G ) ≤ δ ( R ( G ) ) ≤ δ ( G ) + 1 / 2 , δ ( Λ ( G ) ) ≤ δ ( Q ( G ) ) ≤ δ ( Λ ( G ) ) + 1 / 2 , δ ( S ( G ) ) ≤ 2 δ ( R ( G ) ) ≤ δ ( S ( G ) ) + 1 and δ ( R ( G ) ) − 1 / 2 ≤ δ ( Λ ( G ) ) ≤ 5 δ ( R ( G ) ) + 5 / 2 for every graph which is not a tree. Moreover, we also derive some inequalities for the Gromov product and the Gromov product restricted to vertices.

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Reformulated Zagreb Indices of Some Derived Graphs

Mathematics ◽

10.3390/math7040366 ◽

2019 ◽

Vol 7 (4) ◽

pp. 366 ◽

Cited By ~ 6

Author(s):

Jia-Bao Liu ◽

Bahadur Ali ◽

Muhammad Aslam Malik ◽

Hafiz Muhammad Afzal Siddiqui ◽

Muhammad Imran

Keyword(s):

Physical Properties ◽

Chemical Structure ◽

Topological Index ◽

Chemical Reactivity ◽

Line Graph ◽

Total Graph ◽

Zagreb Indices ◽

Subdivision Graph ◽

Chemical Constitution ◽

Vertex Degrees

A topological index is a numeric quantity that is closely related to the chemical constitution to establish the correlation of its chemical structure with chemical reactivity or physical properties. Miličević reformulated the original Zagreb indices in 2004, replacing vertex degrees by edge degrees. In this paper, we established the expressions for the reformulated Zagreb indices of some derived graphs such as a complement, line graph, subdivision graph, edge-semitotal graph, vertex-semitotal graph, total graph, and paraline graph of a graph.

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Line and Subdivision Graphs Determined by T4-Gain Graphs

Mathematics ◽

10.3390/math7100926 ◽

2019 ◽

Vol 7 (10) ◽

pp. 926 ◽

Cited By ~ 3

Author(s):

Abdullah Alazemi ◽

Milica Anđelić ◽

Francesco Belardo ◽

Maurizio Brunetti ◽

Carlos M. da Fonseca

Keyword(s):

Spectral Properties ◽

Incidence Matrix ◽

Multiplicative Group ◽

Line Graph ◽

Characteristic Polynomials ◽

Signed Graphs ◽

Subdivision Graph ◽

Gain Graphs ◽

The Relationship ◽

Gain Graph

Let T 4 = { ± 1 , ± i } be the subgroup of fourth roots of unity inside T , the multiplicative group of complex units. For a T 4 -gain graph Φ = ( Γ , T 4 , φ ) , we introduce gain functions on its line graph L ( Γ ) and on its subdivision graph S ( Γ ) . The corresponding gain graphs L ( Φ ) and S ( Φ ) are defined up to switching equivalence and generalize the analogous constructions for signed graphs. We discuss some spectral properties of these graphs and in particular we establish the relationship between the Laplacian characteristic polynomial of a gain graph Φ , and the adjacency characteristic polynomials of L ( Φ ) and S ( Φ ) . A suitably defined incidence matrix for T 4 -gain graphs plays an important role in this context.

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On the Adjacency, Laplacian, and Signless Laplacian Spectrum of Coalescence of Complete Graphs

Journal of Mathematics ◽

10.1155/2016/5906801 ◽

2016 ◽

Vol 2016 ◽

pp. 1-11 ◽

Cited By ~ 2

Author(s):

S. R. Jog ◽

Raju Kotambari

Keyword(s):

Line Graph ◽

Laplacian Matrix ◽

Spectral Graph Theory ◽

Complete Graphs ◽

Laplacian Spectrum ◽

Spectral Graph ◽

Signless Laplacian Spectrum ◽

Subdivision Graph ◽

Laplacian Energy ◽

Signless Laplacian

Coalescence as one of the operations on a pair of graphs is significant due to its simple form of chromatic polynomial. The adjacency matrix, Laplacian matrix, and signless Laplacian matrix are common matrices usually considered for discussion under spectral graph theory. In this paper, we compute adjacency, Laplacian, and signless Laplacian energy (Qenergy) of coalescence of pair of complete graphs. Also, as an application, we obtain the adjacency energy of subdivision graph and line graph of coalescence from itsQenergy.

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Signless Laplacian Polynomial and Characteristic Polynomial of a Graph

Journal of Discrete Mathematics ◽

10.1155/2013/105624 ◽

2013 ◽

Vol 2013 ◽

pp. 1-4 ◽

Cited By ~ 1

Author(s):

Harishchandra S. Ramane ◽

Shaila B. Gudimani ◽

Sumedha S. Shinde

Keyword(s):

Characteristic Polynomial ◽

Regular Graph ◽

Adjacency Matrix ◽

Line Graph ◽

Induced Subgraphs ◽

Subdivision Graph ◽

Signless Laplacian ◽

The Matrix ◽

Derivatives Of ◽

Degree Matrix

The signless Laplacian polynomial of a graph G is the characteristic polynomial of the matrix Q(G)=D(G)+A(G), where D(G) is the diagonal degree matrix and A(G) is the adjacency matrix of G. In this paper we express the signless Laplacian polynomial in terms of the characteristic polynomial of the induced subgraphs, and, for regular graph, the signless Laplacian polynomial is expressed in terms of the derivatives of the characteristic polynomial. Using this we obtain the characteristic polynomial of line graph and subdivision graph in terms of the characteristic polynomial of induced subgraphs.

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COHERENCE ANALYSIS FOR ITERATED LINE GRAPHS OF MULTI-SUBDIVISION GRAPH

Fractals ◽

10.1142/s0218348x2050067x ◽

2020 ◽

Vol 28 (04) ◽

pp. 2050067

Author(s):

MEIFENG DAI ◽

JIE ZHU ◽

FANG HUANG ◽

YIN LI ◽

LINHE ZHU ◽

...

Keyword(s):

Line Graph ◽

Recursion Formula ◽

Coherence Analysis ◽

Line Graphs ◽

Linear Dynamical System ◽

Laplacian Spectrum ◽

Stochastic Disturbances ◽

Subdivision Graph ◽

Consensus Dynamics ◽

Graph Operation

More and more attention has focused on consensus problem in the study of complex networks. Many researchers investigated consensus dynamics in a linear dynamical system with additive stochastic disturbances. In this paper, we construct iterated line graphs of multi-subdivision graph by applying multi-subdivided-line graph operation. It has been proven that the network coherence can be characterized by the Laplacian spectrum of network. We study the recursion formula of Laplacian eigenvalues of the graphs. After that, we obtain the scalings of the first- and second-order network coherence.

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Distance Degree Index of Some Derived Graphs

Mathematics ◽

10.3390/math7030283 ◽

2019 ◽

Vol 7 (3) ◽

pp. 283 ◽

Cited By ~ 2

Author(s):

Jianzhong Xu ◽

Jia-Bao Liu ◽

Ahsan Bilal ◽

Uzma Ahmad ◽

Hafiz Muhammad Afzal Siddiqui ◽

...

Keyword(s):

Organic Molecules ◽

Line Graph ◽

Chemical Compounds ◽

Topological Indices ◽

Pharmacological Properties ◽

Graph Structure ◽

Total Graph ◽

Graph Vertex ◽

Physical Chemical ◽

Subdivision Graph

Topological indices are numerical values associated with a graph (structure) that can predict many physical, chemical, and pharmacological properties of organic molecules and chemical compounds. The distance degree (DD) index was introduced by Dobrynin and Kochetova in 1994 for characterizing alkanes by an integer. In this paper, we have determined expressions for a DD index of some derived graphs in terms of the parameters of the parent graph. Specifically, we establish expressions for the DD index of a line graph, subdivision graph, vertex-semitotal graph, edge-semitotal graph, total graph, and paraline graph.

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Induced cycle path number of derived graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s179383091650066x ◽

2016 ◽

Vol 08 (04) ◽

pp. 1650066

Author(s):

J. Paulraj Joseph ◽

S. Rosalin

Keyword(s):

Line Graph ◽

Block Graph ◽

Minimum Order ◽

Total Graph ◽

Power Graph ◽

Subdivision Graph ◽

Path Partition ◽

Simple Connected Graph ◽

Path Number ◽

Cycle Path

An induced cycle path partition of a simple connected graph [Formula: see text] is a partition of [Formula: see text] into subsets such that the subsets induce cycles and paths. The minimum order taken over all induced cycle path partitions is called the induced cycle path number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we investigate the induced cycle path number of some derived graphs such as line graph, block graph, subdivision graph, power graph and total graph.

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