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 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.

scholarly journals WIENER INDEX OF INTERVAL WEIGHTED GRAPHS

2021 ◽  
Vol 21 (1) ◽  
pp. 21-28
Author(s):  
SEMIHA BASDAS NURKAHLI ◽  
SERIFE BUYUKKOSE
Keyword(s):  
Topological Index ◽  
Undirected Graph ◽  
Weighted Graph ◽  
Wiener Index ◽  
Weighted Graphs ◽  
Molecular Graphs ◽  
Interval Matrices ◽  
Study Interval

The Wiener index is classic and well-known topological index for the characterization of molecular graphs. In this paper, we study interval weighted graphs such as graphs where the edge weights are interval or interval matrices. In this study firstly we define interval weighted graph. Later we define Wiener index of interval weighted graph. From this definition, we give algorithms for finding Wiener index of interval weighted directed and undirected graph.

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.

Classifying Graphs

2017 ◽  
Author(s):  
Arthur Benjamin ◽  
Gary Chartrand ◽  
Ping Zhang
Keyword(s):  
Graph Theory ◽  
Rational Number ◽  
Regular Graphs ◽  
Weighted Graphs ◽  
Reconstruction Problem ◽  
Regular Polyhedra ◽  
Four Color Theorem ◽  
Stanislaw Ulam ◽  
Irregular Graphs ◽  
Isomorphic Graphs

This chapter considers the richness of mathematics and mathematicians' responses to it, with a particular focus on various types of graphs. It begins with a discussion of theorems from many areas of mathematics that have been judged among the most beautiful, including the Euler Polyhedron Formula; the number of primes is infinite; there are five regular polyhedra; there is no rational number whose square is 2; and the Four Color Theorem. The chapter proceeds by describing regular graphs, irregular graphs, irregular multigraphs and weighted graphs, subgraphs, and isomorphic graphs. It also analyzes the degrees of the vertices of a graph, along with concepts and ideas concerning the structure of graphs. Finally, it revisits a rather mysterious problem in graph theory, introduced by Stanislaw Ulam and Paul J. Kelly, that no one has been able to solve: the Reconstruction Problem.

Composition of Functional Petri Nets

2013 ◽  
pp. 404-464 ◽  
Author(s):  
Dmitry A. Zaitsev
Keyword(s):  
Graph Theory ◽  
Petri Nets ◽  
Petri Net ◽  
Ad Hoc ◽  
Fundamental Equation ◽  
Weighted Graph ◽  
Sequential Composition ◽  
Speed Up ◽  
Networking Protocols ◽  
Linear Invariants

Functional Petri nets and subnets are introduced and studied for the purpose of speed-up of Petri nets analysis with algebraic methods. The authors show that any functional subnet may be generated by a composition of minimal functional subnets. They propose two ways to decompose a Petri net: via logical equations solution and with an ad-hoc algorithm, whose complexity is polynomial. Then properties of functional subnets are studied. The authors show that linear invariants of a Petri net may be computed from invariants of its functional subnets; similar results also hold for the fundamental equation of Petri nets. A technique for Petri nets analysis using composition of functional subnets is also introduced and studied. The authors show that composition-based calculation of invariants and solutions of fundamental equation provides a significant speed-up of computations. For an additional speed-up, they propose a sequential composition of functional subnets. Sequential composition is formalised in the terms of graph theory and was named the optimal collapse of a weighted graph. At last, the authors apply the introduced technique to the analysis of Petri net models of such well-known networking protocols as ECMA, TCP, BGP.

scholarly journals Some New Bounds of Weighted Graph Entropies with GA and Gaurava Indices Edge Weights

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Tiejun Wu ◽  
Hafiz Mutee Ur Rehman ◽  
Yu-Ming Chu ◽  
Deeba Afzal ◽  
Jianfeng Yu
Keyword(s):  
Structural Information ◽  
Weighted Graph ◽  
Shannon’S Entropy ◽  
Molecular Graphs ◽  
Shannon's Entropy ◽  
Complex Graph

Motivated by the concept of Shannon’s entropy, the degree-dependent weighted graph entropy was defined which is now become a tool for measurement of structural information of complex graph networks. The aim of this paper is to study weighted graph entropy. We used GA and Gaurava indices as edge weights to define weighted graph entropy and establish some bounds for different families of graphs. Moreover, we compute the defined weighted entropies for molecular graphs of some dendrimer structures.

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.

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 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.

Some topological indices of dendrimers

2020 ◽  
pp. 2050018
Author(s):  
S. Alyar ◽  
R. Khoeilar ◽  
A. Jahanbani
Keyword(s):  
Graph Theory ◽  
Molecular Graph ◽  
Topological Indices ◽  
Molecular Structures ◽  
Topological Properties ◽  
Zagreb Index ◽  
Zinc Porphyrin ◽  
Molecular Graphs

There are immense applications of graph theory in chemistry and in the study of molecular structures, and after that, it has been increasing exponentially. Molecular graphs have points (vertices) representing atoms and lines (edges) that represent bonds between atoms. In this paper, we study the molecular graph of porphyrin, propyl ether imine, zinc–porphyrin and poly dendrimers and analyzed its topological properties. For this purpose, we have computed topological indices, namely the Albertson index, the sigma index, the Nano-Zagreb index, the first and second hyper [Formula: see text]-indices of porphyrin, propyl ether imine, zinc–porphyrin and poly dendrimers.

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