scholarly journals Line and Subdivision Graphs Determined by T4-Gain Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 926 ◽  
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.

scholarly journals Line graphs of complex unit gain graphs with least eigenvalue -2

2021 ◽  
Vol 37 (37) ◽  
pp. 14-30
Author(s):  
Maurizio Brunetti ◽  
Francesco Belardo
Keyword(s):  
Real Number ◽  
Adjacency Matrix ◽  
Multiplicative Group ◽  
Line Graph ◽  
Line Graphs ◽  
Signed Graphs ◽  
The Real ◽  
Least Eigenvalue ◽  
Gain Graphs ◽  
Gain Graph

Let $\mathbb T$ be the multiplicative group of complex units, and let $\mathcal L (\Phi)$ denote a line graph of a $\mathbb{T}$-gain graph $\Phi$. Similarly to what happens in the context of signed graphs, the real number $\min Spec(A(\mathcal L (\Phi))$, that is, the smallest eigenvalue of the adjacency matrix of $\mathcal L(\Phi)$, is not less than $-2$. The structural conditions on $\Phi$ ensuring that $\min Spec(A(\mathcal L (\Phi))=-2$ are identified. When such conditions are fulfilled, bases of the $-2$-eigenspace are constructed with the aid of the star complement technique.

scholarly journals Local Spectral Properties Under Conjugations

2021 ◽  
Vol 18 (3) ◽  
Author(s):  
Pietro Aiena ◽  
Fabio Burderi ◽  
Salvatore Triolo
Keyword(s):  
Hilbert Space ◽  
Spectral Properties ◽  
Operator Matrices ◽  
Weyl Type Theorems ◽  
Triangular Operator ◽  
Analytic Core ◽  
Complex Symmetric ◽  
Symmetric Operators ◽  
The Relationship

AbstractIn this paper, we study some local spectral properties of operators having form JTJ, where J is a conjugation on a Hilbert space H and $$T\in L(H)$$ T ∈ L ( H ) . We also study the relationship between the quasi-nilpotent part of the adjoint $$T^*$$ T ∗ and the analytic core K(T) in the case of decomposable complex symmetric operators. In the last part we consider Weyl type theorems for triangular operator matrices for which one of the entries has form JTJ, or has form $$JT^*J$$ J T ∗ J . The theory is exemplified in some concrete cases.

ON THE NORMALISED LAPLACIAN SPECTRUM, DEGREE-KIRCHHOFF INDEX AND SPANNING TREES OF GRAPHS

2015 ◽  
Vol 91 (3) ◽  
pp. 353-367 ◽  
Author(s):  
JING HUANG ◽  
SHUCHAO LI
Keyword(s):  
Lower Bound ◽  
Characteristic Polynomial ◽  
Regular Graph ◽  
Spanning Trees ◽  
Line Graph ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Subdivision Graph ◽  
Number Of Spanning Trees ◽  
Sharp Lower Bound

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 $q(G)$).

scholarly journals Rainfall Variation: Implication on Cocoa Yield in Ondo State, Nigeria

2018 ◽  
pp. 16-25
Keyword(s):  
Tropical Rainforest ◽  
Line Graph ◽  
Yield Data ◽  
Rainfall Variation ◽  
Cash Crops ◽  
Ondo State ◽  
R Value ◽  
The Relationship ◽  
Cocoa Plantation ◽  
Cocoa Yield

Cocoa is known as one of the notable cultivated cash crops of the tropical rainforest of the world that is rain dependent. The study examines the effect of rainfall variation on the yield of cocoa plantation in Ondo State, Nigeria. Data used for the study includes the rainfall data of 15 years from 2000 to 2014 collected from Ondo state agro climatological office as well as cocoa yield data for the same period of time from Ondo State ministry of agriculture and forest resources. Descriptive statistical method was employed to determine the relationship between both variables in which the result shows direct relationship between rainfall and cocoa yield. Results were presented using bar charts and line graph for the time series analysis of the variables. Linear regression statistical analysis was used to predict cocoa yield with certain amount of rainfall with the correlation coefficient ‘r’ value of 0.97 which implies that rainfall changes go a long way to determine the same variation trend in the cocoa yield. Though, not only the quantity of rainfall within the range of rainfall required for the growth of this crop affect the yield but its distribution. A little millimeter of rainfall above or below the required range of rain for cocoa plantation greatly affects cocoa yield.

scholarly journals Local Spectral Theory for R and S Satisfying RnSRn = Rj

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 120
Author(s):  
Salvatore Triolo
Keyword(s):  
Spectral Theory ◽  
Spectral Properties ◽  
Operator Equations ◽  
The Relationship ◽  
Local Spectral Theory

In this paper, we analyze local spectral properties of operators R,S and RS which satisfy the operator equations RnSRn=Rj and SnRSn=Sj for same integers j≥n≥0. We also continue to study the relationship between the local spectral properties of an operator R and the local spectral properties of S. Thus, we investigate the transmission of some local spectral properties from R to S and we illustrate our results with an example. The theory is exemplified in some cases.

Voice Onset Time in Consonant Cluster Errors: Can Phonetic Accommodation Differentiate Cognitive From Motor Errors?

2014 ◽  
Vol 57 (5) ◽  
pp. 1577-1588 ◽  
Author(s):  
Marianne Pouplier ◽  
Stefania Marin ◽  
Susanne Waltl
Keyword(s):  
Phonological Processing ◽  
Spectral Properties ◽  
Voice Onset Time ◽  
Onset Time ◽  
Consonant Cluster ◽  
Acoustic Data ◽  
Processing Level ◽  
Production Models ◽  
Planning Processes ◽  
The Relationship

Purpose Phonetic accommodation in speech errors has traditionally been used to identify the processing level at which an error has occurred. Recent studies have challenged the view that noncanonical productions may solely be due to phonetic, not phonological, processing irregularities, as previously assumed. The authors of the present study investigated the relationship between phonological and phonetic planning processes on the basis of voice onset time (VOT) behavior in consonant cluster errors. Method Acoustic data from 22 German speakers were recorded while eliciting errors on sibilant-stop clusters. Analyses consider VOT duration as well as intensity and spectral properties of the sibilant. Results Of all incorrect responses, 28% failed to show accommodation. Sibilant intensity and spectral properties differed from correct responses irrespective of whether VOT was accommodated. Conclusions The data overall do not allow using (a lack of) accommodation as a diagnostic as to the processing level at which an error has occurred. The data support speech production models that allow for an integrated view of phonological and phonetic processing.

scholarly journals Hyperbolicity on Graph Operators

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 360 ◽  
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.

Laplace and Extended Adjacency Matrices for Isomorphism Detection of Kinematic Chains Using the Characteristic Polynomial Approach

2005 ◽  
Author(s):  
Rajesh Pavan Sunkari ◽  
Linda C. Schmidt
Keyword(s):  
Characteristic Polynomial ◽  
Spectral Properties ◽  
Kinematic Chain ◽  
Detection Algorithm ◽  
Isomorphism Problem ◽  
Characteristic Polynomials ◽  
Kinematic Chains ◽  
Laplace Matrix ◽  
Adjacency Matrices ◽  
Single Matrix

The kinematic chain isomorphism problem is one of the most challenging problems facing mechanism researchers. Methods using the spectral properties, characteristic polynomial and eigenvectors, of the graph related matrices were developed in literature for isomorphism detection. Detection of isomorphism using only the spectral properties corresponds to a polynomial time isomorphism detection algorithm. However, most of the methods used are either computationally inefficient or unreliable (i.e., failing to identify non-isomorphic chains). This work establishes the reliability of using the characteristic polynomial of the Laplace matrix for isomorphism detection of a kinematic chain. The Laplace matrix of a graph is used extensively in the field of algebraic graph theory for characterizing a graph using its spectral properties. The reliability in isomorphism detection of the characteristic polynomial of the Laplace matrix was comparable with that of the adjacency matrix. However, using the characteristic polynomials of both the matrices is superior to using either method alone. In search for a single matrix whose characteristic polynomial unfailingly detects isomorphism, novel matrices called the extended adjacency matrices are developed. The reliability of the characteristic polynomials of these matrices is established. One of the proposed extended adjacency matrices is shown to be the best graph matrix for isomorphism detection using the characteristic polynomial approach.

Binomial incidence matrix of a semigraph

2020 ◽  
pp. 2150017
Author(s):  
Jyoti Shetty ◽  
G. Sudhakara
Keyword(s):  
Graph Theory ◽  
Complete Graph ◽  
Adjacency Matrix ◽  
Incidence Matrix ◽  
Systematic Approach ◽  
Line Graph ◽  
The Matrix ◽  
Theory Of Graphs

A semigraph, defined as a generalization of graph by  Sampathkumar, allows an edge to have more than two vertices. The idea of multiple vertices on edges gives rise to multiplicity in every concept in the theory of graphs when generalized to semigraphs. In this paper, we define a representing matrix of a semigraph [Formula: see text] and call it binomial incidence matrix of the semigraph [Formula: see text]. This matrix, which becomes the well-known incidence matrix when the semigraph is a graph, represents the semigraph uniquely, up to isomorphism. We characterize this matrix and derive some results on the rank of the matrix. We also show that a matrix derived from the binomial incidence matrix satisfies a result in graph theory which relates incidence matrix of a graph and adjacency matrix of its line graph. We extend the concept of “twin vertices” in the theory of graphs to semigraph theory, and characterize them. Finally, we derive a systematic approach to show that the binomial incidence matrix of any semigraph on [Formula: see text] vertices can be obtained from the incidence matrix of the complete graph [Formula: see text].

scholarly journals Reformulated Zagreb Indices of Some Derived Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 366 ◽  
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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