International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 53712, 3 pages

On the basis number of the corona of graphs

Mohammad Shakhatreh and Ahmad Al-Rhayyel

Department of Mathematics, Yarmouk University, Irbid 211-63, Jordan

Received 5 February 2005; Revised 19 June 2006; Accepted 22 June 2006

Copyright © 2006 Mohammad Shakhatreh and Ahmad Al-Rhayyel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The basis number b(G) of a graph G is defined to be the least integer k such that G has a k-fold basis for its cycle space. In this note, we determine the basis number of the corona of graphs, in fact we prove that b(vT)=2 for any tree and any vertex v not in T, b(vH)b(H)+2, where H is any graph and v is not a vertex of H, also we prove that if G=G1G2 is the corona of two graphs G1 and G2, then b(G1)b(G)max{b(G1),b(G2)+2}, moreover we prove that if G is a Hamiltonian graph, then b(vG)b(G)+1, where v is any vertex not in G, and finally we give a sequence of remarks which gives the basis number of the corona of some of special graphs.