On the basis number and the minimum cycle bases of the wreath product of some graphs i

 Access to full text

 Full (PDF)

1. [1] M. Anderson and M. Lipman, The wreath product of graphs, in: Graphs and Applications (Boulder, Colo., 1982), (Wiley-Intersci. Publ., Wiley, New York, 1985) 23-39. 
2. [2] A.A. Ali, The basis number of complete multipartite graphs, Ars Combin. 28 (1989) 41-49. Zbl0728.05058
3. [3] A.A. Ali, The basis number of the direct product of paths and cycles, Ars Combin. 27 (1989) 155-163. 
4. [4] A.A. Ali and G.T. Marougi, The basis number of cartesian product of some graphs, J. Indian Math. Soc. 58 (1992) 123-134. Zbl0880.05055
5. [5] A.S. Alsardary and J. Wojciechowski, The basis number of the powers of the complete graph, Discrete Math. 188 (1998) 13-25, doi: 10.1016/S0012-365X(97)00271-9. Zbl0958.05074
6. [6] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (America Elsevier Publishing Co. Inc., New York, 1976). Zbl1226.05083
7. [7] W.-K. Chen, On vector spaces associated with a graph, SIAM J. Appl. Math. 20 (1971) 525-529, doi: 10.1137/0120054. 
8. [8] D.M. Chickering, D. Geiger and D. HecKerman, On finding a cycle basis with a shortest maximal cycle, Inform. Process. Lett. 54 (1994) 55-58, doi: 10.1016/0020-0190(94)00231-M. Zbl0875.68685
9. [9] L.O. Chua and L. Chen, On optimally sparse cycles and coboundary basis for a linear graph, IEEE Trans. Circuit Theory 20 (1973) 54-76. 
10. [10] G.M. Downs, V.J. Gillet, J.D. Holliday and M.F. Lynch, Review of ring perception algorithms for chemical graphs, J. Chem. Inf. Comput. Sci. 29 (1989) 172-187, doi: 10.1021/ci00063a007. 
11. [11] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition (Wiley, New York, 2000). 
12. [12] W. Imrich and P. Stadler, Minimum cycle bases of product graphs, Australas. J. Combin. 26 (2002) 233-244. Zbl1009.05078
13. [13] M.M.M. Jaradat, On the basis number of the direct product of graphs, Australas. J. Combin. 27 (2003) 293-306. Zbl1021.05060
14. [14] M.M.M. Jaradat, The basis number of the direct product of a theta graph and a path, Ars Combin. 75 (2005) 105-111. Zbl1074.05051
15. [15] M.M.M. Jaradat, An upper bound of the basis number of the strong product of graphs, Discuss. Math. Graph Theory 25 (2005) 391-406, doi: 10.7151/dmgt.1291. Zbl1107.05049
16. [16] M.M.M. Jaradat, M.Y. Alzoubi and E.A. Rawashdeh, The basis number of the Lexicographic product of different ladders, SUT J. Math. 40 (2004) 91-101. Zbl1072.05049
17. [17] A. Kaveh, Structural Mechanics, Graph and Matrix Methods. Research Studies Press (Exeter, UK, 1992). Zbl0858.73002
18. [18] G. Liu, On connectivities of tree graphs, J. Graph Theory 12 (1988) 435-459, doi: 10.1002/jgt.3190120318. Zbl0649.05044
19. [19] S. MacLane, A combinatorial condition for planar graphs, Fundamenta Math. 28 (1937) 22-32. Zbl0015.37501
20. [20] M. Plotkin, Mathematical basis of ring-finding algorithms in CIDS, J. Chem. Doc. 11 (1971) 60-63, doi: 10.1021/c160040a013. 
21. [21] E.F. Schmeichel, The basis number of a graph, J. Combin. Theory (B) 30 (1981) 123-129, doi: 10.1016/0095-8956(81)90057-5. Zbl0385.05031
22. [22] P. Vismara, Union of all the minimum cycle bases of a graph, Electr. J. Combin. 4 (1997) 73-87. Zbl0885.05101
23. [23] D.J.A. Welsh, Kruskal's theorem for matroids, Proc. Cambridge Phil, Soc. 64 (1968) 3-4, doi: 10.1017/S030500410004247X. Zbl0157.55302

1. M.M.M. Jaradat, Minimal cycle bases of the lexicographic product of graphs