scholarly journals Efficient Algorithm for Generating Maximal L-Reflexive Trees

Symmetry ◽  
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.

scholarly journals Nonplanarity of Iterated Line Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Jing Wang
Keyword(s):  
Line Graph ◽  
Crossing Number ◽  
Line Graphs ◽  
Full Characterization ◽  
Iterated Line Graph

The 1-crossing index of a graph G is the smallest integer k such that the k th iterated line graph of G has crossing number greater than 1. In this paper, we show that the 1-crossing index of a graph is either infinite or it is at most 5. Moreover, we give a full characterization of all graphs with respect to their 1-crossing index.

On Spanning and Dominating Circuits in Graphs

1977 ◽  
Vol 20 (2) ◽  
pp. 215-220 ◽  
Author(s):  
L. Lesniak-Foster ◽  
James E. Williamson
Keyword(s):  
Line Graph ◽  
Dominating Set ◽  
Line Graphs ◽  
Sufficient Condition ◽  
Vertex Set ◽  
On Line ◽  
One To One

AbstractA set E of edges of a graph G is said to be a dominating set of edges if every edge of G either belongs to E or is adjacent to an edge of E. If the subgraph 〈E〉 induced by E is a trail T, then T is called a dominating trail of G. Dominating circuits are defined analogously. A sufficient condition is given for a graph to possess a spanning (and thus dominating) circuit and a sufficient condition is given for a graph to possess a spanning (and thus dominating) trail between each pair of distinct vertices. The line graph L(G) of a graph G is defined to be that graph whose vertex set can be put in one-to-one correspondence with the edge set of G in such a way that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. The existence of dominating trails and circuits is employed to present results on line graphs and second iterated line graphs, respectively.

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

1990 ◽  
Vol 112 (1) ◽  
pp. 79-83 ◽  
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.

scholarly journals Line graph of unit graphs associated with finite commutative rings

2021 ◽  
Author(s):  
Pranjali ◽  
Amit Kumar ◽  
Pooja Sharma
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Line Graph ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Common Vertex ◽  
Unit Graph ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Nonzero Identity

For a given graph G, its line graph denoted by L(G) is a graph whose vertex set V (L(G)) = E(G) and {e1, e2} ∈ E(L(G)) if e1 and e2 are incident to a common vertex in G. Let R be a finite commutative ring with nonzero identity and G(R) denotes the unit graph associated with R. In this manuscript, we have studied the line graph L(G(R)) of unit graph G(R)  associated with R. In the course of the investigation, several basic properties, viz., diameter, girth, clique, and chromatic number of L(G(R)) have been determined. Further, we have derived sufficient conditions for L(G(R)) to be Planar and Hamiltonian

scholarly journals Line Graph Associated to Graph of a Near-Ring with Respect to an Ideal

2021 ◽  
Vol 52 ◽  
Author(s):  
Moytri Sarmah
Keyword(s):  
Line Graph ◽  
Dominating Set ◽  
Clique Number ◽  
Common Vertex ◽  
Vertex Set

Let N be a near-ring and I be an ideal of N. The graph of N with respect to I is a graph with V (N ) as vertex set and any two distinct vertices x and y are adjacent if and only if xNy ⊆ I oryNx ⊆ I. This graph is denoted by GI(N). We define the line graph of GI(N) as a graph with each edge of GI (N ) as vertex and any two distinct vertices are adjacent if and only if their corresponding edges share a common vertex in the graph GI (N ). We denote this graph by L(GI (N )). We have discussed the diameter, girth, clique number, dominating set of L(GI(N)). We have also found conditions for the graph L(GI(N)) to be acycle graph.

BALANCED UNITARY CAYLEY SIGRAPHS OVER FINITE COMMUTATIVE RINGS

2014 ◽  
Vol 13 (05) ◽  
pp. 1350152 ◽  
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.

scholarly journals Planarity and outerplanarity indexes of the unit, unitary and total graphs

Filomat ◽  
2017 ◽  
Vol 31 (9) ◽  
pp. 2827-2836
Author(s):  
Zahra Barati
Keyword(s):  
Commutative Ring ◽  
Line Graphs ◽  
Full Characterization ◽  
Finite Commutative Ring

In this paper, we consider the problem of planarity and outerplanarity of iterated line graphs of the unit, unitary and total graphs when R is a finite commutative ring. We give a full characterization of all these graphs with respect to their planarity and outerplanarity indexes.

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

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Akbar Ali ◽  
Atif Nadeem ◽  
Zahid Raza ◽  
Wael W. Mohammed ◽  
Elsayed M. Elsayed
Keyword(s):  
Line Graph ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Unicyclic Graphs ◽  
Maximum Value ◽  
Vertex Set ◽  
Fixed Order ◽  
Unique Graph

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.

Multidecompositions of line graphs of complete graphs

2019 ◽  
Vol 11 (03) ◽  
pp. 1950035 ◽  
Author(s):  
S. Ganesamurthy ◽  
P. Paulraja ◽  
R. Srimathi
Keyword(s):  
Necessary Conditions ◽  
Line Graph ◽  
Common Vertex ◽  
Line Graphs ◽  
Complete Graphs

By a [Formula: see text]-decomposition of a graph [Formula: see text] we mean a decomposition of [Formula: see text] into [Formula: see text] copies of [Formula: see text] [Formula: see text] copies of [Formula: see text] and [Formula: see text] copies of [Formula: see text], where [Formula: see text] are non-negative integers. In this paper, it is proved that the necessary conditions for the existence of a [Formula: see text]-decomposition of [Formula: see text] are sufficient, where B, the bowtie, is the graph with two triangles having exactly one common vertex. Further, it is shown that the line graph of [Formula: see text] [Formula: see text] has a [Formula: see text]-decomposition except for [Formula: see text].

A Thorough Theoretical Exploration of Intriguing Characteristics of Cyclo[18]carbon: Geometry, Bonding Nature, Aromaticity, Weak Interaction, Reactivity, Excited States, Vibrations, Molecular Dynamics and Various Molecular Properties

2019 ◽  
Author(s):  
Tian Lu ◽  
Qinxue Chen ◽  
Zeyu Liu
Keyword(s):  
Molecular Dynamics ◽  
Excited States ◽  
Electronic Excitation ◽  
Ring Structure ◽  
Molecular Properties ◽  
Molecular Vibration ◽  
Full Characterization ◽  
Long Time ◽  
Almost All

Although cyclo[18]carbon has been theoretically and experimentally investigated since long time ago, only very recently it was prepared and directly observed by means of STM/AFM in condensed phase (Kaiser et al., <i>Science</i>, <b>365</b>, 1299 (2019)). The unique ring structure and dual 18-center π delocalization feature bring a variety of unusual characteristics and properties to the cyclo[18]carbon, which are quite worth to be explored. In this work, we present an extremely comprehensive and detailed investigation on almost all aspects of the cyclo[18]carbon, including (1) Geometric characteristics (2) Bonding nature (3) Electron delocalization and aromaticity (4) Intermolecular interaction (5) Reactivity (6) Electronic excitation and UV/Vis spectrum (7) Molecular vibration and IR/Raman spectrum (8) Molecular dynamics (9) Response to external field (10) Electron ionization, affinity and accompanied process (11) Various molecular properties. We believe that our full characterization of the cyclo[18]carbon will greatly deepen researchers' understanding of this system, and thereby help them to utilize it in practice and design its various valuable derivatives.

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