Variable Neighborhood Search for Extremal Graphs. 10. Comparison of Irregularity Indices for Chemical Trees

Gutman, Iván; Hansen, Pierre; Mélot, Hadrien

doi:10.1021/ci0342775

Cited by 59 publications

(36 citation statements)

References 22 publications

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“…The irregularity in graphs is a topic of interest for the analysis of models based on them, as exemplified by the study of acyclic molecules [15]. Over the recent years, the literature has shown a significant number of works devoted to the subject, usually dealing, with a single exception, with measures calculated using only the degree sequences.…”

Section: The Purpose Of the Discussionmentioning

confidence: 99%

Irregularidad de grafos: Discusión, extensiones de grafos y nuevas propuestas

Netto

2015

RMTA

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This paper presents some measures of graph irregularity found in the literature. From their discussion two important points appear: first, the absence of relationship between all of them, but a single exception, with the structures of the corresponding graphs and, moreover, their known extremal values correspond to graphs having degree sequences with few different values. Two new measures are proposed, seeking to meet these points. Their values are calculated for extremal graphs associated with other measures and for antiregular graphs. Finally, we calculate the boxplots of all these measures for some sets of graphs taken from the literature and also for four sets where the ordered degree sequences are constant. All measures involved have polynomial complexity.Keywords: Graphs; irregularity; irregularity measures; graph extensions. ResumenEste artículo presenta un análisis de las medidas de irregularidad de grafos que se encuentran en la literatura. Desde su discusión dos puntos importantes aparecen: primero, la ausencia de relación entre todos ellos, sino una sola excepción -con las estructuras de los grafos correspondientes y, además, sus valores extremales conocidos corresponden a grafos que tienen secuencias de grados con pocos valores diferentes. Se proponen dos nuevas medidas, tratando de cumplir con estos puntos. Sus valores se calculan para grafos extremales asociados con otras medidas y para grafos antiregulares. Por último, se determinan los gráficos de quartiles o boxplots de todas estas medidas, para algunos conjuntos de grafos de la literatura y para cuatro conjuntos donde las secuencias ordenadas de grados son constantes. Todas las medidas estudiadas tienen complejidad polinómica.

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Section: The Purpose Of the Discussionmentioning

confidence: 99%

Irregularidad de grafos: Discusión, extensiones de grafos y nuevas propuestas

Netto

2015

RMTA

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“…The well-known result, in spectral graph theory, λ 1 (G) ≥ d(G) with equality if and only if G is a regular graph, was proved by Collatz and Sinogowitz (1957). Then, they proposed to consider the difference between the index and the average degree as a measure of the irregularity of a graph [other definitions of irregularity in graphs have been proposed, see Alberston (1997) and Bell (1992), and for a comparison between them see Gutman et al (2005)]. Thus, the irregularity of a graph G is defined by…”

Section: The Irregularitymentioning

confidence: 99%

“…Applications to graph theory are given in Aouchiche et al (2001Aouchiche et al ( , 2006Aouchiche et al ( ,b, 2008aAouchiche et al ( ,b, 2009aAouchiche et al ( ,b,c, 2010, ,b, 2009), Belhaiza et al (2005, Cvetković et al (2001), , Sedlar et al (2008), and Stevanović et al (2008); applications to chemical graph theory in , Caporossi et al (1999a,b), Fowler et al (2001), Gutman et al (2005), Hansen and Mélot (2003), ; and developments of these results in Caporossi et al (2003) and Gutman et al (1999).…”

Section: Introductionmentioning

confidence: 99%

Open problems on graph eigenvalues studied with AutoGraphiX

Aouchiche

Caporossi

Hansen

2013

EURO Journal on Computational Optimization

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Since the late forties of the last century, methods of operations research have been extensively used to solve problems in graph theory, and graph theory has been extensively used to model operations research problems and to solve optimization problems on graphs, e.g., shortest paths and network flow problems. More recently, methods of operations research and artificial intelligence have been used to advance graph theory per se, i.e., to find conjectures on graph theory invariants, to refute such conjectures and in some cases to find automated proofs or ideas of proofs. Among other systems, the AutoGraphiX system was developed since 1997 at GERAD (Montreal) by the present authors. Extensive experiments have been conducted which led to 1,700 conjectures, about 800 of which turned out to be easy and could be proved by the system, and about 600 further ones were proved by hand by us or graph theorists from various countries. Moreover, these results led to many generalizations and further papers. In this paper, we study four theoretical problems related to the eigenvalues of (the adjacency matrix of) a connected graph and to which AutoGraphiX was applied. Three of the problems are related to the maximum value of the irregularity, the maximum spectral spread and the upper bound of Nordhaus-Gaddum type on the index, over the class of connected graphs on n vertices. The fourth problem concerns the maximization of the energy (the sum of the absolute values of the eigenvalues) of a connected graph with fixed numbers of vertices and of cycles. We present a brief survey of the papers on or in connection with these problems, and give some new partial results.

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“…The majority of irregularity indices belong to the family of degree-based graph invariants, but there exist eigenvalue-based irregularity indices as well [1,3,4,7,8,12,15,21,24,26,27].…”

Section: Introduction Preliminary Considerationsmentioning

confidence: 99%

On the construction and comparison of graph irregularity indices

Réti¹,

Tóth-Laufer²

2017

Kragujevac J Sci

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ABSTRACT. Irregularity indices are generally used for quantitative characterization of topological structure of non-regular graphs. According to a widely accepted preconception, using a topological invariant (called a graph irregularity index) for that purpose, the results of graph irregularity classification should be consistent with our subjective judgements (intuitive feeling). In the case of structurally strongly similar graphs, it is difficult to select the proper irregularity index by which the irregularity ranking (ordering) of graphs can be performed in a consistent manner to our preliminary expectations. In this work we investigate various possibilities of constructing so-called composite irregularity indices obtained as a function of traditional irregularity indices and test their discriminatory performance. Moreover, it has been demonstrated in examples, that in some cases the subjective evaluation of graph irregularities leads to false conclusions. This phenomenon is called the paradox of quantitative graph irregularity characterization. From our study we have concluded that results of graph irregularity measuring depend not only on the choice of irregularity indices but it is influenced strongly on the preselected set of graphs to be investigated. Similar problems arise for the quantitative evaluation of information content, complexity or branching of molecular graphs. 54

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Variable Neighborhood Search for Extremal Graphs. 10. Comparison of Irregularity Indices for Chemical Trees

Cited by 59 publications

References 22 publications

Irregularidad de grafos: Discusión, extensiones de grafos y nuevas propuestas

Irregularidad de grafos: Discusión, extensiones de grafos y nuevas propuestas

Open problems on graph eigenvalues studied with AutoGraphiX

On the construction and comparison of graph irregularity indices

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