scholarly journals Characteristic polynomials of some weighted graph bundles and its application to links

1994 ◽  
Vol 17 (3) ◽  
pp. 503-510 ◽  
Author(s):  
Moo Young Sohn ◽  
Jaeun Lee
Keyword(s):  
Characteristic Polynomial ◽  
Weighted Graph ◽  
Double Covering ◽  
Characteristic Polynomials ◽  
Weighted Graphs

In this paper, we introduce weighted graph bundles and study their characteristic polynomial. In particular, we show that the characteristic polynomial of a weightedK2(K¯2)-bundles over a weighted graphG?can be expressed as a product of characteristic polynomials two weighted graphs whose underlying graphs areGAs an application, we compute the signature of a link whose corresponding weighted graph is a double covering of that of a given link.

Graph Theory in Chemistry II. Graph-Theoretical Description of Heteroconjugated Molecules

1975 ◽  
Vol 30 (12) ◽  
pp. 1696-1699 ◽  
Author(s):  
A. Graovac ◽  
O. E. Polansky ◽  
N. Trinajstić ◽  
N. Tyutyulkov
Keyword(s):  
Graph Theory ◽  
Characteristic Polynomial ◽  
Weighted Graph ◽  
Theoretical Description ◽  
Weighted Graphs ◽  
Conjugated Molecules ◽  
Molecular Graphs

Abstract The Sachs' formula which relates the structure of a (vertex-and edge) -weighted graph and its characteristic polynomial is given. Weighted graphs are used to represent conjugated molecules containing the variety of heteroatoms and heterobonds. Vertices and edges in such molecular graphs have different weights following the deviation of the corresponding (Hiickel) Coulomb and resonance integrals from the standard (benzene) values.

scholarly journals Spread of Epidemic Disease on Edge-Weighted Graphs from a Database: A Case Study of COVID-19

2021 ◽  
Vol 18 (9) ◽  
pp. 4432
Author(s):  
Ronald Manríquez ◽  
Camilo Guerrero-Nancuante ◽  
Felipe Martínez ◽  
Carla Taramasco
Keyword(s):  
Public Health ◽  
Infectious Diseases ◽  
Preventive Measures ◽  
Weighted Graph ◽  
Real Data ◽  
Sir Model ◽  
Weighted Graphs ◽  
Epidemic Disease ◽  
The Real

The understanding of infectious diseases is a priority in the field of public health. This has generated the inclusion of several disciplines and tools that allow for analyzing the dissemination of infectious diseases. The aim of this manuscript is to model the spreading of a disease in a population that is registered in a database. From this database, we obtain an edge-weighted graph. The spreading was modeled with the classic SIR model. The model proposed with edge-weighted graph allows for identifying the most important variables in the dissemination of epidemics. Moreover, a deterministic approximation is provided. With database COVID-19 from a city in Chile, we analyzed our model with relationship variables between people. We obtained a graph with 3866 vertices and 6,841,470 edges. We fitted the curve of the real data and we have done some simulations on the obtained graph. Our model is adjusted to the spread of the disease. The model proposed with edge-weighted graph allows for identifying the most important variables in the dissemination of epidemics, in this case with real data of COVID-19. This valuable information allows us to also include/understand the networks of dissemination of epidemics diseases as well as the implementation of preventive measures of public health. These findings are important in COVID-19’s pandemic context.

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.

WEIGHTED-EDGE-COLORING OF k-DEGENERATE GRAPHS AND BIN-PACKING

2011 ◽  
Vol 12 (01n02) ◽  
pp. 109-124
Author(s):  
FLORIAN HUC
Keyword(s):  
Bin Packing ◽  
Edge Coloring ◽  
Weighted Graph ◽  
Packing Problem ◽  
Bipartite Graphs ◽  
Specific Class ◽  
Weighted Graphs ◽  
Outerplanar Graphs ◽  
Underlying Graph ◽  
Multiple Edges

The weighted-edge-coloring problem of an edge-weighted graph whose weights are between 0 and 1, consists in finding a coloring using as few colors as possible and satisfying the following constraints: the sum of weights of edges with the same color and incident to the same vertex must be at most 1. In 1991, Chung and Ross conjectured that if G is bipartite, then [Formula: see text] colors are always sufficient to weighted-edge-color (G,w), where [Formula: see text] is the maximum of the sums of the weights of the edges incident to a vertex. We prove this is true for edge-weighted graphs with multiple edges whose underlying graph is a tree. We further generalise this conjecture to non-bipartite graphs and prove the generalised conjecture for simple edge-weighted outerplanar graphs. Finally, we introduce a list version of this coloring together with the list-bin-packing problem, which allows us to obtain new results concerning the original coloring for a specific class of graphs, namely the k-weight-degenerate weighted graph.

scholarly journals On the class of square Petrie matrices induced by cyclic permutations

2004 ◽  
Vol 2004 (31) ◽  
pp. 1617-1622
Author(s):  
Bau-Sen Du
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Topological Entropy ◽  
Characteristic Polynomial ◽  
Characteristic Polynomials ◽  
Dynamical Properties

Letn≥2be an integer and letP={1,2,…,n,n+1}. LetZpdenote the finite field{0,1,2,…,p−1}, wherep≥2is a prime. Then every mapσonPdetermines a realn×nPetrie matrixAσwhich is known to contain information on the dynamical properties such as topological entropy and the Artin-Mazur zeta function of the linearization ofσ. In this paper, we show that ifσis acyclicpermutation onP, then all such matricesAσare similar to one another overZ2(but not overZpfor any primep≥3) and their characteristic polynomials overZ2are all equal to∑k=0nxk. As a consequence, we obtain that ifσis acyclicpermutation onP, then the coefficients of the characteristic polynomial ofAσare all odd integers and hence nonzero.

Isoperimetric properties of weighted graphs related to the Laplacian spectrum and canonical correlations

2002 ◽  
Vol 39 (3-4) ◽  
pp. 425-441 ◽  
Author(s):  
M. Bolla ◽  
G. Molnár-Sáska
Keyword(s):  
Volume Ratio ◽  
Weighted Graph ◽  
Weighted Graphs ◽  
Laplacian Spectrum ◽  
Minimum Requirement ◽  
Cheeger Constant ◽  
Canonical Correlations ◽  
Laplacian Spectra ◽  
Weighted Laplacian ◽  
Similar Cluster

The relation between isoperimetric properties and Laplacian spectra of weighted graphs is investigated. The vertices are classified into k clusters with „few" inter-cluster edges of „small" weights (area) and „similar" cluster sizes (volumes). For k=2 the Cheeger constant represents the minimum requirement for the area/volume ratio and it is estimated from above by v?1(2-?1), where ?1 is the smallest positive eigenvalue of the weighted Laplacian. For k?2 we define the k-density of a weighted graph that is a generalization of the Cheeger constant and estimated from below by Si=1k-1?i and from above by c2 Si=1k-1 ?i, where 0<?1=…=Sk-1 are the smallest Laplacian eigenvalues and the constant c?1 depends on the metric classification properties of the corresponding eigenvectors. Laplacian spectra are also related to canonical correlations in a probabilistic setup.

scholarly journals Graphs Having Most of Their Eigenvalues Shared by a Vertex Deleted Subgraph

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1663
Author(s):  
Alexander Farrugia
Keyword(s):  
Characteristic Polynomial ◽  
Simple Graph ◽  
Characteristic Polynomials ◽  
Reconstruction Problem ◽  
Vertex Set ◽  
Disconnected Graphs ◽  
Polynomial Reconstruction ◽  
Standing Problem

Let G be a simple graph and {1,2,…,n} be its vertex set. The polynomial reconstruction problem asks the question: given a deck P(G) containing the n characteristic polynomials of the vertex deleted subgraphs G−1, G−2, …, G−n of G, can ϕ(G,x), the characteristic polynomial of G, be reconstructed uniquely? To date, this long-standing problem has only been solved in the affirmative for some specific classes of graphs. We prove that if there exists a vertex v such that more than half of the eigenvalues of G are shared with those of G−v, then this fact is recognizable from P(G), which allows the reconstruction of ϕ(G,x). To accomplish this, we make use of determinants of certain walk matrices of G. Our main result is used, in particular, to prove that the reconstruction of the characteristic polynomial from P(G) is possible for a large subclass of disconnected graphs, strengthening a result by Sciriha and Formosa.

scholarly journals Constant Factor Approximation for Tracking Paths and Fault Tolerant Feedback Vertex Set

2021 ◽  
pp. 23-38
Author(s):  
Václav Blažej ◽  
Pratibha Choudhary ◽  
Dušan Knop ◽  
Jan Matyáš Křišt’an ◽  
Ondřej Suchý ◽  
...  
Keyword(s):  
Approximation Algorithm ◽  
Fault Tolerant ◽  
Minimum Weight ◽  
Weighted Graph ◽  
Constant Factor ◽  
Weighted Graphs ◽  
Feedback Vertex Set ◽  
Fixed Integer ◽  
Vertex Set

AbstractConsider a vertex-weighted graph G with a source s and a target t. Tracking Paths requires finding a minimum weight set of vertices (trackers) such that the sequence of trackers in each path from s to t is unique. In this work, we derive a factor 66-approximation algorithm for Tracking Paths in weighted graphs and a factor 4-approximation algorithm if the input is unweighted. This is the first constant factor approximation for this problem. While doing so, we also study approximation of the closely related r-Fault Tolerant Feedback Vertex Set problem. There, for a fixed integer r and a given vertex-weighted graph G, the task is to find a minimum weight set of vertices intersecting every cycle of G in at least $$r+1$$ r + 1 vertices. We give a factor $$\mathcal {O}(r^2)$$ O ( r 2 ) approximation algorithm for r-Fault Tolerant Feedback Vertex Set if r is a constant.

Clustering Vertices in Weighted Graphs

2011 ◽  
pp. 285-298
Author(s):  
Derry Tanti Wijaya ◽  
Stephane Bressan
Keyword(s):  
Graph Algorithms ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Weighted Graph ◽  
Graph Clustering ◽  
Weighted Graphs ◽  
Similarity Relations ◽  
Natural Notion ◽  
Cluster Density ◽  
Markov Clustering

Clustering is the unsupervised process of discovering natural clusters so that objects within the same cluster are similar and objects from different clusters are dissimilar. In clustering, if similarity relations between objects are represented as a simple, weighted graph where objects are vertices and similarities between objects are weights of edges; clustering reduces to the problem of graph clustering. A natural notion of graph clustering is the separation of sparsely connected dense sub graphs from each other based on the notion of intra-cluster density vs. inter-cluster sparseness. In this chapter, we overview existing graph algorithms for clustering vertices in weighted graphs: Minimum Spanning Tree (MST) clustering, Markov clustering, and Star clustering. This includes the variants of Star clustering, MST clustering and Ricochet.

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