scholarly journals On the oriented incidence energy and decomposable graphs

Filomat ◽  
2009 ◽  
Vol 23 (3) ◽  
pp. 243-249 ◽  
Author(s):  
Dragan Stevanovic ◽  
Abreu de ◽  
Freitas de ◽  
Cybele Vinagre ◽  
Renata Del-Vecchio
Keyword(s):  
Incidence Matrix ◽  
Oriented Graph ◽  
Singular Values ◽  
Simple Graph ◽  
Arbitrary Orientation ◽  
Decomposable Graphs

Let G be a simple graph with n vertices and m edges. Let edges of G be given an arbitrary orientation, and let Q be the vertex-edge incidence matrix of such oriented graph. The oriented incidence energy of G is then the sum of singular values of Q. We show that for any n 2 N, there exists a set of n graphs with O(n) vertices having equal oriented incidence energy.

scholarly journals Oriented incidence energy and threshold graphs

Filomat ◽  
2011 ◽  
Vol 25 (2) ◽  
pp. 1-8 ◽  
Author(s):  
Dragan Stevanovic
Keyword(s):  
Incidence Matrix ◽  
Oriented Graph ◽  
Singular Values ◽  
Simple Graph ◽  
Arbitrary Orientation ◽  
Threshold Graphs

Let G be a simple graph with n vertices and m edges. Let edges of G be given an arbitrary orientation, and let Q be the vertex-edge incidence matrix of such oriented graph. The oriented incidence energy of G is then the sum of singular values of Q. We show that for any n?9, there exists at least ([n/9]/2)+1 distinct pairs of graphs on n vertices having equal oriented incidence energy.

scholarly journals Bounds for Incidence Energy of Some Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Weizhong Wang ◽  
Dong Yang
Keyword(s):  
Regular Graph ◽  
Upper Bound ◽  
Maximum Degree ◽  
Incidence Matrix ◽  
Line Graph ◽  
Singular Values ◽  
Simple Graph

LetGbe a simple graph. The incidence energy (IEfor short) ofGis defined as the sum of the singular values of the incidence matrix. In this paper, a new upper bound forIEof graphs in terms of the maximum degree is given. Meanwhile, bounds forIEof the line graph of a semiregular graph and the paraline graph of a regular graph are obtained.

scholarly journals On the Incidence Energy of Some Toroidal Lattices

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Jia-Bao Liu ◽  
Jinde Cao ◽  
Jin Xie
Keyword(s):  
Closed Form ◽  
Incidence Matrix ◽  
Singular Values ◽  
Chemical Physics ◽  
Analysis Method ◽  
Kagome Lattice ◽  
Kagomé Lattice ◽  
Asymptotic Values

The incidence energyIE(G), defined as the sum of the singular values of the incidence matrix ofG, is a much studied quantity with well known applications in chemical physics. In this paper, we derived the closed-form formulae expressing the incidence energy of the 3.12.12 lattice, triangular kagomé lattice, andS(m,n)lattice, respectively. Simultaneously, the explicit asymptotic values of the incidence energy in these lattices are obtained by utilizing the applications of analysis method with the help of software calculation.

scholarly journals Chromatic Polynomials of Oriented Graphs

10.37236/8240 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Danielle Cox ◽  
Christopher Duffy
Keyword(s):  
Polynomial Time ◽  
Open Problem ◽  
Oriented Graph ◽  
Chromatic Polynomial ◽  
Simple Graph ◽  
Interval Graphs ◽  
Chromatic Polynomials ◽  
Oriented Graphs ◽  
Real Roots ◽  
The Family

The oriented chromatic polynomial of a oriented graph outputs the number of oriented $k$-colourings for any input $k$. We fully classify those oriented graphs for which the oriented graph has the same chromatic polynomial as the underlying simple graph, closing an open problem posed by Sopena. We find that such oriented graphs can be both identified and constructed in polynomial time as they are exactly the family of quasi-transitive oriented co-interval graphs. We study the analytic properties of this polynomial and show that there exist oriented graphs which have chromatic polynomials have roots, including negative real roots,  that cannot be realized as the root of any chromatic polynomial of a simple graph.

scholarly journals On oriented graphs with minimal skew energy

2014 ◽  
Vol 27 ◽  
Author(s):  
Shi-Cai Gong ◽  
Xueliang Li ◽  
Guanghui Xu
Keyword(s):  
Adjacency Matrix ◽  
Variable Neighborhood Search ◽  
Integral Formula ◽  
Oriented Graph ◽  
Singular Values ◽  
Neighborhood Search ◽  
External Energy ◽  
Oriented Graphs ◽  
Skew Energy ◽  
Skew Adjacency Matrix

Let S(G^σ) be the skew-adjacency matrix of an oriented graph Gσ. The skew energy of G^σ is the sum of all singular values of its skew-adjacency matrix S(G^σ). This paper first establishes an integral formula for the skew energy of an oriented graph. Then, it determines all oriented graphs with minimal skew energy among all connected oriented graphs on n vertices with m (n ≤ m < 2(n − 2)) arcs, which is analogous to the conjecture for the energy of undirected graphs proposed by Caporossi et al. [G. Caporossi, D. Cvetkovic, I. Gutman, and P. Hansen. Variable neighborhood search for extremal graphs. 2. Finding graphs with external energy. J. Chem. Inf. Comput. Sci., 39:984–996, 1999].

scholarly journals Spectrum of large random Markov chains: Heavy-tailed weights on the oriented complete graph

2017 ◽  
Vol 06 (02) ◽  
pp. 1750006 ◽  
Author(s):  
Charles Bordenave ◽  
Pietro Caputo ◽  
Djalil Chafaï ◽  
Daniele Piras
Keyword(s):  
Probability Measure ◽  
Domain Of Attraction ◽  
Oriented Graph ◽  
Singular Values ◽  
Empirical Measure ◽  
Stable Law ◽  
Smoothness Assumption ◽  
Infinite Tree ◽  
Heavy Tailed ◽  
First Moment

We consider the random Markov matrix obtained by assigning i.i.d. non-negative weights to each edge of the complete oriented graph. In this study, the weights have unbounded first moment and belong to the domain of attraction of an alpha-stable law. We prove that as the dimension tends to infinity, the empirical measure of the singular values tends to a probability measure which depends only on alpha, characterized as the expected value of the spectral measure at the root of a weighted random tree. The latter is a generalized two-stage version of the Poisson weighted infinite tree (PWIT) introduced by David Aldous. Under an additional smoothness assumption, we show that the empirical measure of the eigenvalues tends to a non-degenerate isotropic probability measure depending only on alpha and supported on the unit disk of the complex plane. We conjecture that the limiting support is actually formed by a strictly smaller disk.

scholarly journals Symbolic powers of vertex cover ideals

2020 ◽  
Vol 30 (06) ◽  
pp. 1167-1183
Author(s):  
S. Selvaraja
Keyword(s):  
Polynomial Ring ◽  
Vertex Cover ◽  
Simple Graph ◽  
Symbolic Powers ◽  
Vertex Decomposable ◽  
Decomposable Graphs ◽  
Linear Quotients

Let [Formula: see text] be a finite simple graph and [Formula: see text] denote its vertex cover ideal in a polynomial ring over a field [Formula: see text]. In this paper, we show that all symbolic powers of vertex cover ideals of certain vertex-decomposable graphs have linear quotients. Using these results, we give various conditions on a subset [Formula: see text] of the vertices of [Formula: see text] so that all symbolic powers of vertex cover ideals of [Formula: see text], obtained from [Formula: see text] by adding a whisker to each vertex in [Formula: see text], have linear quotients. For instance, if [Formula: see text] is a vertex cover of [Formula: see text], then all symbolic powers of [Formula: see text] have linear quotients. Moreover, we compute the Castelnuovo–Mumford regularity of symbolic powers of certain vertex cover ideals.

A SVD and Modified Firefly Optimization Based Robust Digital Image Watermarking Technique

2017 ◽  
Vol 7 (7) ◽  
pp. 268
Author(s):  
Chauhan Usha ◽  
Singh Rajeev Kumar
Keyword(s):  
Digital Media ◽  
Firefly Algorithm ◽  
Image Watermarking ◽  
Singular Values ◽  
Robust Watermarking ◽  
Scaling Factors ◽  
Geometric Attacks ◽  
Host Image ◽  
Firefly Optimization ◽  
Value Decomposition

Digital Watermarking is a technology, to facilitate the authentication, copyright protection and Security of digital media. The objective of developing a robust watermarking technique is to incorporate the maximum possible robustness without compromising with the transparency. Singular Value Decomposition (SVD) using Firefly Algorithm provides this objective of an optimal robust watermarking technique. Multiple scaling factors are used to embed the watermark image into the host by multiplying these scaling factors with the Singular Values (SV) of the host image. Firefly Algorithm is used to optimize the modified host image to achieve the highest possible robustness and transparency. This approach can significantly increase the quality of watermarked image and provide more robustness to the embedded watermark against various attacks such as noise, geometric attacks, filtering attacks etc.

On Some Properties of Semi-rings

2018 ◽  
pp. 35-50
Author(s):  
A. I. Belousov
Keyword(s):  
Linear Equations ◽  
Oriented Graph ◽  
Theoretical Computer Science ◽  
Alternative Methods ◽  
Main Role ◽  
Cost Matrix ◽  
Sequential Elimination ◽  
The Matrix ◽  
Systems Of Linear Equations ◽  
The Cost

The main objective of this paper is to prove a theorem according to which a method of successive elimination of unknowns in the solution of systems of linear equations in the semi-rings with iteration gives the really smallest solution of the system. The proof is based on the graph interpretation of the system and establishes a relationship between the method of sequential elimination of unknowns and the method for calculating a cost matrix of a labeled oriented graph using the method of sequential calculation of cost matrices following the paths of increasing ranks. Along with that, and in terms of preparing for the proof of the main theorem, we consider the following important properties of the closed semi-rings and semi-rings with iteration.We prove the properties of an infinite sum (a supremum of the sequence in natural ordering of an idempotent semi-ring). In particular, the proof of the continuity of the addition operation is much simpler than in the known issues, which is the basis for the well-known algorithm for solving a linear equation in a semi-ring with iteration.Next, we prove a theorem on the closeness of semi-rings with iteration with respect to solutions of the systems of linear equations. We also give a detailed proof of the theorem of the cost matrix of an oriented graph labeled above a semi-ring as an iteration of the matrix of arc labels.The concept of an automaton over a semi-ring is introduced, which, unlike the usual labeled oriented graph, has a distinguished "final" vertex with a zero out-degree.All of the foregoing provides a basis for the proof of the main theorem, in which the concept of an automaton over a semi-ring plays the main role.The article's results are scientifically and methodologically valuable. The proposed proof of the main theorem allows us to relate two alternative methods for calculating the cost matrix of a labeled oriented graph, and the proposed proofs of already known statements can be useful in presenting the elements of the theory of semi-rings that plays an important role in mathematical studies of students majoring in software technologies and theoretical computer science.

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