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Revista de Matemática Teoría y Aplicaciones

versão impressa ISSN 1409-2433

Rev. Mat vol.22 no.2 San José Jul./Dez. 2015

 

Articles

Graph irregularity: discussion, graph extensions and new proposals

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

Paulo Oswaldo Boaventura-Netto 1  

1Programa de Engenharia de Produção, COPPE, Federal University of Rio de Janeiro, Brazil. E-Mail: boaventu@po.coppe.ufrj.br

Abstract

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

Resumen

Este 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.

Palabras clave: grafos; irregularidad; medidas de irregularidad; extensiones de grafos

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1Mathematics Subject Classification: 05C07, 05C35, 05C99.

Received: March 30, 2014; Revised: April 07, 2015; Accepted: May 21, 2015

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