Characterization of Decentralized and Quotient Fixed Modes via Graph Theory

Graph Theory International audience In this document, we study the scope of the following graph model: each vertex is assigned to a box in ℝd and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in ℝ. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., rooted directed path). We also study the particular case where each representative element is the center of its respective box. In this particular case, we provide constructive representations for interval, block and outerplanar graphs. Finally, we show that the general and the particular model are not equivalent by constructing a graph family that separates the two cases.

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Protein Interactions for Functional Genomics

International Journal of Knowledge Discovery in Bioinformatics ◽

10.4018/ijkdb.2012100102 ◽

2012 ◽

Vol 3 (4) ◽

pp. 15-30

Author(s):

Pablo Minguez ◽

Joaquin Dopazo

Keyword(s):

Graph Theory ◽

Protein Interactions ◽

Biological Networks ◽

State Of The Art ◽

Protein Protein Interactions ◽

Pairwise Interactions ◽

Novel Approach ◽

Functional Profiling ◽

Available Resources

Here the authors review the state of the art in the use of protein-protein interactions (ppis) within the context of the interpretation of genomic experiments. They report the available resources and methodologies used to create a curated compilation of ppis introducing a novel approach to filter interactions. Special attention is paid in the complexity of the topology of the networks formed by proteins (nodes) and pairwise interactions (edges). These networks can be studied using graph theory and a brief introduction to the characterization of biological networks and definitions of the more used network parameters is also given. Also a report on the available resources to perform different modes of functional profiling using ppi data is provided along with a discussion on the approaches that have typically been applied into this context. They also introduce a novel methodology for the evaluation of networks and some examples of its application.

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Finite Characterization of the Coarsest Balanced Coloring of a Network

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420502120 ◽

2020 ◽

Vol 30 (14) ◽

pp. 2050212

Author(s):

Ian Stewart

Keyword(s):

Graph Theory ◽

Adjacency Matrix ◽

Universal Cover ◽

Running Time ◽

Set Up ◽

Balanced Coloring ◽

Refinement Algorithm ◽

Partition Refinement

Balanced colorings of networks correspond to flow-invariant synchrony spaces. It is known that the coarsest balanced coloring is equivalent to nodes having isomorphic infinite input trees, but this condition is not algorithmic. We provide an algorithmic characterization: two nodes have the same color for the coarsest balanced coloring if and only if their [Formula: see text]th input trees are isomorphic, where [Formula: see text] is the number of nodes. Here [Formula: see text] is the best possible. The proof is analogous to that of Leighton’s theorem in graph theory, using the universal cover of the network and the notion of a symbolic adjacency matrix to set up a partition refinement algorithm whose output is the coarsest balanced coloring. The running time of the algorithm is cubic in [Formula: see text].

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Closed orders and closed graphs

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.1515/auom-2016-0034 ◽

2016 ◽

Vol 24 (2) ◽

pp. 159-167

Author(s):

Marilena Crupi

Keyword(s):

Graph Theory ◽

Interval Graph ◽

Linear Ordering ◽

Proper Interval Graph

Abstract The class of closed graphs by a linear ordering on their sets of vertices is investigated. A recent characterization of such a class of graphs is analyzed by using tools from the proper interval graph theory.

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Characterization of the spontaneous electroencephalographic activity in Alzheimer's disease using disequilibria and graph theory

2013 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC) ◽

10.1109/embc.2013.6610917 ◽

2013 ◽

Cited By ~ 1

Author(s):

Jesus Poza ◽

Maria Garcia ◽

Carlos Gomez ◽

Alejandro Bachiller ◽

Alicia Carreres ◽

...

Keyword(s):

Alzheimer’S Disease ◽

Alzheimer's Disease ◽

Graph Theory ◽

Electroencephalographic Activity

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A Characterization of Almost-Planar Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548300002005 ◽

1996 ◽

Vol 5 (3) ◽

pp. 227-245 ◽

Cited By ~ 4

Author(s):

Bradley S. Gubser

Keyword(s):

Graph Theory ◽

Planar Graph ◽

Planar Graphs ◽

Explicit Description ◽

Famous Result ◽

Minor Criterion ◽

Excluded Minor

Kuratowski's Theorem, perhaps the most famous result in graph theory, states that K5 and K3,3 are the only non-planar graphs for which both G\e, the deletion of the edge e, and G/e, the contraction of the edge e, are planar for all edges e of G. We characterize the almost-planar graphs, those non-planar graphs for which G\e or G/e is planar for all edges e of G. This paper gives two characterizations of the almost-planar graphs: an explicit description of the structure of almost-planar graphs; and an excluded minor criterion. We also give a best possible bound on the number of edges of an almost-planar graph.

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Study of θϕ Networks via Zagreb Connection Indices

Symmetry ◽

10.3390/sym13111991 ◽

2021 ◽

Vol 13 (11) ◽

pp. 1991

Author(s):

Muhammad Asif ◽

Bartłomiej Kizielewicz ◽

Atiq ur Rehman ◽

Muhammad Hussain ◽

Wojciech Sałabun

Keyword(s):

Graph Theory ◽

Predictive Power ◽

Interconnection Network ◽

Construction Process ◽

Acentric Factor ◽

Network Systems ◽

Topological Characterization ◽

Chemical Graph Theory ◽

Initial Graph

Graph theory can be used to optimize interconnection network systems. The compatibility of such networks mainly depends on their topology. Topological indices may characterize the topology of such networks. In this work, we studied a symmetric network θϕ formed by ϕ time repetition of the process of joining θ copies of a selected graph Ω in such a way that corresponding vertices of Ω in all the copies are joined with each other by a new edge. The symmetry of θϕ is ensured by the involvement of complete graph Kθ in the construction process. The free hand to choose an initial graph Ω and formation of chemical graphs using θϕΩ enhance its importance as a family of graphs which covers all the pre-defined graphs, along with space for new graphs, possibly formed in this way. We used Zagreb connection indices for the characterization of θϕΩ. These indices have gained worth in the field of chemical graph theory in very small duration due to their predictive power for enthalpy, entropy, and acentric factor. These computations are mathematically novel and assist in topological characterization of θϕΩ to enable its emerging use.

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Dynamic Characterization of Typical Electrical Circuits via Structural Properties

Mathematical Problems in Engineering ◽

10.1155/2016/7870462 ◽

2016 ◽

Vol 2016 ◽

pp. 1-13 ◽

Cited By ~ 8

Author(s):

Sofía Avila-Becerril ◽

Gerardo Espinosa-Pérez ◽

Paul Fernandez

Keyword(s):

Graph Theory ◽

Steady State ◽

Structural Properties ◽

Mesh Networks ◽

Dynamic Characterization ◽

State Behavior ◽

Electrical Circuits ◽

Mathematical Properties ◽

Steady State Behavior

The characterization of a class of electrical circuits is carried out in terms of both stability properties and steady-state behavior. The main contribution is the interpretation of the electrical topology (how the elements that conform the circuits are interconnected) in terms of mathematical properties derived from the structure of their models. In this sense, at what extent the topology by itself defines the dynamic behavior of the systems is explained. The study is based on the graph theory allowing capturing, departing from the well-known Kirchhoff laws, the topology of the circuits into several matrices with specific structure. The algebraic analysis of these matrices permits identifying conditions that determine whether the system is stable in the sense of Lyapunov and the kind of steady-state behavior that it exhibits. The approach is mainly focused on typical topologies widely used in practice, namely, radial, ring, and mesh networks.

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