1Universidad de Los Andes, Mérida, Venezuela. Email: juanrada@ula.ve
]]> Abstract
A directed graph G is nonderogatory if its adjacency matrix A is nonderogatory, i.e., the characteristic polynomial of A is equal to the minimal polynomial of A. Given integers r≥ 2 and h≥ 3, a directed windmill Mh(r) is a directed graph obtained by coalescing r dicycles of length h in one vertex. In this article we solve a conjecture proposed by Gan and Koo ([3]): Mh(r) is nonderogatory if and only if r=2.
Key words: Nonderogatory matrix, characteristic polynomial of directedgraphs, directed windmills.
Un grafo dirigido G es no-derogatorio si su matriz de adyacencia A es no-derogatoria, es decir el polinomio característico de A es igual al polinomio minimal de A. Dados enteros r≥ 2 y h≥ 3, el molino de viento dirigido Mh(r) es un grafo dirigido que se obtiene por medio de la coalescencia de r diciclos de longitud h en un vértice. En este artículo resolvemos una conjetura propuesta por Gan y Koo ([3]): Mh(r) es no-derogatorio si, y sólo si, r=2.
Palabras clave: matriz no-derogatoria, polinomio característico de grafos dirigidos, molinos de viento dirigidos.
Texto completo disponible en PDF
References
[1] Cvetkovic, D., Doob, M. & Sachs, H., Spectra of graphs, Academic Press, New York, 1980. [ Links ]
[2] Gan, C., Some results on annihilatingly unique digraphs, M.Sc. Thesis, Univesity of Malaya, 1995. Kuala Lumpur, Malaysia.. [ Links ]
[3] Gan, C. & Koo, V., On annihilating uniqueness of directed windmills, `Proceedings of the ATCM´, (2002), ATCM. Melaka, Malaysia.. [ Links ]
[4] Lam, K., On digraphs with unique annihilating polynomial, Ph.D. Thesis, University of Malaya, 1990. Kuala Lumpur, Malaysia.. [ Links ]
[5] Lam, K. & Lim, C., `The characteristic polynomial of ladder digraph and an annihilating uniqueness theorem´, Discrete Mathematics 151, (1996), 161-167. [ Links ]
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCMv42n1a05,
AUTHOR = {Rada, Juan},
TITLE = {{Nonderogatory directed windmills}},
JOURNAL = {Revista Colombiana de Matemáticas},
YEAR = {2008}, ]]>
volume = {42},
number = {1},
pages = {61-66}
}