Duplication Of Vertex Research Articles

A minimal free resolution of the duplication of a monomial ideal

This article explores a novel operation of a monomial ideal, termed duplication, which produces a monomial ideal I⋄\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I^\\diamond $$\\end{document} from another I. This operation is inspired by and generalizes how vertex duplication affects the edge ideal of a graph. The main result describes a multigraded minimal free resolution of I⋄\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I^\\diamond $$\\end{document} as long as we know a resolution for I. Consequently, we obtain formulas for the projective dimension and depth of I⋄\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I^\\diamond $$\\end{document} in terms of the corresponding invariants of I.

Face Bimagic Mean Labeling of Herschel Graphs and Its Operations

A bijection ϕ: V(G)→{1,2, . . . , |V(G)|} is called a (1,0,0)-F-face bimagic mean labeling (FBMML) of G if the induced face labeling ϕ*(fi) = = = k1 or k2, constant. This work investigates the possibility of face bimagic mean labeling of a Herschel graph. Also, a study of face bimagic mean labeling during the process of fusion of vertices of degree 3 of a Herschel graph and the duplication of vertices by the edge of a Herschel graph is provided in this research paper.

Genetic networks encode secrets of their past

Research shows that gene duplication followed by either repurposing or removal of duplicated genes is an important contributor to evolution of gene and protein interaction networks. We aim to identify which characteristics of a network can arise through this process, and which must have been produced in a different way. To model the network evolution, we postulate vertex duplication and edge deletion as evolutionary operations on graphs. Using the novel concept of an ancestrally distinguished subgraph, we show how features of present-day networks require certain features of their ancestors. In particular, ancestrally distinguished subgraphs cannot be introduced by vertex duplication. Additionally, if vertex duplication and edge deletion are the only evolutionary mechanisms, then a graph’s ancestrally distinguished subgraphs must be contained in all of the graph’s ancestors. We analyze two experimentally derived genetic networks and show that our results accurately predict lack of large ancestrally distinguished subgraphs, despite this feature being statistically improbable in associated random networks. This observation is consistent with the hypothesis that these networks evolved primarily via vertex duplication. The tools we provide open the door for analyzing ancestral networks using current networks. Our results apply to edge-labeled (e.g. signed) graphs which are either undirected or directed.

Some results on strong 2 - vertex duplication self switching of some connected graphs

A vertex \(v \in V(G)\) is said to be a self vertex switching of \(G\) if \(G\) is isomorphic to \(G^v\), where \(G^v\) is the graph obtained from \(G\) by deleting all edges of \(G\) incident to \(v\) in \(G\) and adding all edges incident to \(v\) which are not in \(G\). A vertex \(v^{\prime}\) is the duplication of \(v\) if all the vertices which are adjacent to \(v\) in \(G\) are also adjacent to \(v^{\prime}\) in \(D(v G)\), which is the duplication graph of \(G\). Duplication self vertex switching of various graphs are given in the literature. In this paper we discuss about the 2-vertex duplication self switching graphs.

The split-and-drift random graph, a null model for speciation

We introduce a new random graph model motivated by biological questions relating to speciation. This random graph is defined as the stationary distribution of a Markov chain on the space of graphs on {1,…,n}. The dynamics of this Markov chain is governed by two types of events: vertex duplication, where at constant rate a pair of vertices is sampled uniformly and one of these vertices loses its incident edges and is rewired to the other vertex and its neighbors; and edge removal, where each edge disappears at constant rate. Besides the number of vertices n, the model has a single parameter rn.Using a coalescent approach, we obtain explicit formulas for the first moments of several graph invariants such as the number of edges or the number of complete subgraphs of order k. These are then used to identify five non-trivial regimes depending on the asymptotics of the parameter rn. We derive an explicit expression for the degree distribution, and show that under appropriate rescaling it converges to classical distributions when the number of vertices goes to infinity. Finally, we give asymptotic bounds for the number of connected components, and show that in the sparse regime the number of edges is Poissonian.

NEMo: An Evolutionary Model With Modularity for PPI Networks.

Modeling the evolution of biological networks is a major challenge. Biological networks are usually represented as graphs; evolutionary events not only include addition and removal of vertices and edges but also duplication of vertices and their associated edges. Since duplication is viewed as a primary driver of genomic evolution, recent work has focused on duplication-based models. Missing from these models is any embodiment of modularity, a widely accepted attribute of biological networks. Some models spontaneously generate modular structures, but none is known to maintain and evolve them. We describe network evolution with modularity (NEMo), a new model that embodies modularity. NEMo allows modules to appear and disappear and to fission and to merge, all driven by the underlying edge-level events using a duplication-based process. We also introduce measures to compare biological networks in terms of their modular structure; we present comparisons between NEMo and existing duplication-based models and run our measuring tools on both generated and published networks.

Gracefulness of graphs obtained from vertex duplication

Gracefulness of graphs obtained from vertex duplication

Some spectral properties of cographs

A graph containing no path of length three as an induced subgraph is called a cograph. In this article, we give a recursive definition of cographs in terms of the vertex duplication and co-duplication operations. We then establish that no cographs have eigenvalues in the interval (−1,0), generalizing the same known result for threshold graphs. As a consequence, we present combinatorial descriptions for the multiplicities of 0 and −1 as eigenvalues of cographs. This provides a short proof of a known result.

B-Chromatic number of some wheel related graphs

A proper coloring $f$ is a $b$-coloring of the vertices of graph $G$ such that in each color class there exists a vertex that has neighbours in every other color classes. The $b$-chromatic number $\varphi(G)$ of a graph $G$ is the largest integer $k$ for which $G$ admits a $b$-coloring with $k$ colors. If $\chi(G)$ is the chromatic number of $G$ and $b$-coloring exists for every integer $k$ satisfying the inequality $\chi(G) \leq k \leq \varphi(G)$ then $G$ is called $b$-continuous. The $b$-spectrum $S_b(G)$ of a graph $G$ is the set of $k$ integers(colors) for which $G$ has a $b$-coloring. We investigate $b$-chromatic number for the graphs obtained from wheel $W_n$ by means of duplication of vertices. We also discuss $b$-continuity and $b$-spectrum for such graphs.

Cylindrical and Toroidal Parameterizations Without Vertex Seams

Abstract A simple rendering method to avoid vertex seams in cylindrical and toroidal UV mappings used for texture mapping is presented. (A vertex seam is a vertex duplication of a polygonal mesh with different texture coordinates assigned to the two geometrically coinciding copies.) As a result, the method leads to simpler, leaner, replication-free data structures. Is also allows for a higher degree of proceduralism in generation of texture coordinates. The method is general, trivial to implement (exhaustive pseudocode is provided), very low in cost on resources (with a virtually null impact on performance), and it leverages only basic mechanisms widely available in most GPU implementations. An open-source implementation is available at the address provided at the end of this article.

Some New Results on Domination Integrity of Graphs

The domination integrity of a connected graph G= (V(G), E(G)) is denoted as DI(G) and defined by DI(G) = min{*S*+ m(G-S) : S is a dominating set } where m(G-S) is the order of a maximum component of G-S . We discuss domination integrity in the context of some graph operations like duplication of an edge by vertex and duplication of vertex by an edge.

Strongly Multiplicative Labeling for Some Cycle Related Graphs

In this paper we discuss strongly multiplicative labeling of some cycle related graphs. We prove that cycle with one chord, cycle with twin chords and cycle with triangle are strongly multiplicative graphs. In addition to this we prove that the graphs obtained by fusion and duplication of vertices in cycle admit strongly multiplicative labeling.

New Benchmarks for Large-Scale Networks with Given Maximum Degree and Diameter

Large-scale networks have become ubiquitous elements of our society. Modern social networks, supported by communication and travel technology, have grown in size and complexity to unprecedented scales. Computer networks, such as the Internet, have a fundamental impact on commerce, politics and culture. The study of networks is also central in biology, chemistry and other natural sciences. Unifying aspects of these networks are a small maximum degree and a small diameter, which are also shared by many network models, such as small-world networks. Graph theoretical methodologies can be instrumental in the challenging task of predicting, constructing and studying the properties of large-scale networks. This task is now necessitated by the vulnerability of large networks to phenomena such as cross-continental spread of disease and botnets (networks of malware). In this article, we produce the new largest known networks of maximum degree 17 ≤ Δ ≤ 20 and diameter 2 ≤ D ≤ 10, using a wide range of techniques and concepts, such as graph compounding, vertex duplication, Kronecker product, polarity graphs and voltage graphs. In this way, we provide new benchmarks for networks with given maximum degree and diameter, and a complete overview of state-of-the-art methodology that can be used to construct such networks.

Some New Large Compound Graphs

This paper deals with some new constructions of large (Δ, D) graphs, i.e., graphs with maximum degree Δ and diameter D and many vertices. Most constructions presented here are based on the compound graphs technique. The basic idea of compound graphs consists of connecting together several copies of a given graph according to the structure of another one.Using a variation on this classical theme, we build some new large graphs and some new large bipartite graphs. We will also apply the method of vertex duplication to improve some entries in the tables of known large bipartite graphs.

On the unilateral (?,D*)-problem

Large digraphs of a specified maximum degree and unilateral diameter are given for small values of these parameters. The constructions are based on different techniques such as voltage digraphs, digraph products, join of cycles, and vertex duplication. Finally, a table with the results is given.