International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 53712, 3 pages
doi:10.1155/IJMMS/2006/53712
On the basis number of the corona of graphs
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.
Abstract
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(v∘T)=2 for any tree and
any vertex v not in T, b(v∘H)≤b(H)+2, where H is any graph and v is not a vertex of H, also we prove that if
G=G1∘G2 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(v∘G)≤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.