A combinatorial problem on a directed graph

Marrero, Osvaldo; Pasles, PaulC.; Marrero, Osvaldo; Pasles, PaulC.

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Revista de Matemática Teoría y Aplicaciones

Print version ISSN 1409-2433

Rev. Mat vol.23 n.2 San José Jul./Dec. 2016

Artículos

A combinatorial problem on a directed graph

Un problema combinatorio basado en un grafo orientado

Osvaldo Marrero¹

PaulC. Pasles²

^¹Department of Mathematics and Statistics, Villanova University, Villanova, Pennsylvania, U.S.A. E-Mail: Osvaldo.Marrero@villanova.edu

^²Department of Mathematics and Statistics, Villanova University, Villanova, Pennsylvania, U.S.A. E-Mail: Paul.Pasles@villanova.edu

Abstract

We consider two options for a particle's entire journey through a certain directed graph. Both options involve a random assignment to the journey route to be followed. We are interested in the option that offers, on average, the shortest route. Therefore, we determine the average journey length for each of the two options. As part of our analysis, we prove some combinatorial identities that appear to be new. Some suggestions for further work are given.

Keywords: directed graphs; games on graphs; combinatorial identities; combinatorial probability

Resumen

Se consideran dos opciones para la jornada total de una partícula que se desplaza a través de un cierto grafo orientado. Bajo ambas opciones, la ruta de la jornada es asignada aleatoriamente. Nos interesa saber la opción bajo la cual uno espera la ruta más corta. Por eso, para cada opción, determinamos la esperanza matemática del largo de la ruta. Al parecer novedosas, algunas identidades combinatorias son demostradas como parte de nuestro análisis. Para concluir, mencionamos varias oportunidades para futuros estudios.

Palabras clave: grafos orientados; juegos en grafos; identidades combinatorias; probabilidad combinatoria

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References

Chartrand, G.; Lesniak, L.; Zhang, P.(2010) Graphs & Digraphs, 5th edition. CRC Press, Boca Raton FL. [ Links ]

Grimmett, G.; Welsh, D.(2014) Probability: An Introduction, 2d edition. Oxford University Press, Oxford, United Kingdom. [ Links ]

Received: May 13, 2015; Revised: April 14, 2016; Accepted: April 22, 2016

This is an open-access article distributed under the terms of the Creative Commons Attribution License