Total edge irregularity strength of toroidal fullerene. (English) Zbl 1306.05213
Summary: A toroidal fullerene (toroidal polyhex) is a cubic bipartite graph embedded on the torus such that each face is a hexagon. An edge irregular total \(k\)-labeling of a graph \(G\) is such a labeling of the vertices and edges with labels \(1, 2,\dots, k\) that the weights of any two different edges are distinct, where the weight of an edge is the sum of the label of the edge itself and the labels of its two endvertices. The minimum \(k\) for which the graph \(G\) has an edge irregular total \(k\)-labeling is called the total edge irregularity strength, \(\mathrm{tes}(G)\). In this paper we determine the exact value of the total edge irregularity strength of toroidal polyhexes.
MSC:
| 05C78 | Graph labelling (graceful graphs, bandwidth, etc.) |
| 05C90 | Applications of graph theory |
| 93E10 | Estimation and detection in stochastic control theory |
Keywords:
fullerene; toroidal polyhex; total edge irregularity strength; edge irregular total labelingReferences:
| [1] | Ahmad, A., Bača, M., Siddiqui, M.K.: On edge irregular total labeling of categorical product of two cycles. Theory Comput. Syst. (2013). doi: 10.1007/s00224-013-9470-3 |
| [2] | Bača M., Jendroľ S., Miller M., Ryan J.: On irregular total labellings. Discrete Math. 307, 1378–1388 (2007) · Zbl 1115.05079 · doi:10.1016/j.disc.2005.11.075 |
| [3] | Beuerle F., Herrmann C., Whalley A.C., Valente C., Gamburd A., Ratner M.A., Stoddart J.F.: Optical and vibrational properties of toroidal carbon nanotubes. Chem. Eur. J. 17, 3868–3875 (2011) · doi:10.1002/chem.201002758 |
| [4] | Brandt S., Miškuf J., Rautenbach D.: On a conjecture about edge irregular total labellings. J. Graph Theory. 57, 333–343 (2008) · Zbl 1188.05109 · doi:10.1002/jgt.20287 |
| [5] | Cash G.G.: Simple means of computing the Kekulé structure count for toroidal polyhex fullerenes. J. Chem. Inf. Comput. Sci. 38, 58–61 (1998) · doi:10.1021/ci970057d |
| [6] | Chunling T., Xiaohui L., Yuansheng Y., Liping W.: Irregular total labellings of C m C n . Utilitas Math. 81, 3–13 (2010) · Zbl 1207.05175 |
| [7] | Deza M., Fowler P.W., Rassat A., Rogers K.M.: Fullerenes as tilings of surfaces. J. Chem. Inf. Comput. Sci. 40, 550–558 (2000) · doi:10.1021/ci990066h |
| [8] | Haque K.M.M.: Irregular total labellings of generalized Petersen graphs. Theory Comput. Syst. 50, 537–544 (2012) · Zbl 1254.05172 · doi:10.1007/s00224-011-9350-7 |
| [9] | Ivančo J., Jendroľ S.: Total edge irregularity strength of trees. Discussiones Math. Graph Theory 26, 449–456 (2006) · Zbl 1135.05066 · doi:10.7151/dmgt.1337 |
| [10] | Jendroǐ S., Miškuf J., Soták R.: Total edge irregularity strength of complete and complete bipartite graphs. Electron. Notes Discrete Math. 28, 281–285 (2007) · Zbl 1291.05175 · doi:10.1016/j.endm.2007.01.041 |
| [11] | Jendroǐ S., Miškuf J., Soták R.: Total edge irregularity strength of complete graphs and complete bipartite graphs. Discrete Math. 310, 400–407 (2010) · Zbl 1216.05131 · doi:10.1016/j.disc.2009.03.006 |
| [12] | Kang M.H.: Toroidal fullerenes with the Cayley graph structures. Discrete Math. 311, 2384–2395 (2011) · Zbl 1238.05068 · doi:10.1016/j.disc.2011.06.018 |
| [13] | Kirby E.C., Mallion R.B., Pollak P.: Toridal polyhexes. J. Chem. Soc. Faraday Trans. 89(12), 1945–1953 (1993) · doi:10.1039/ft9938901945 |
| [14] | Kirby E.C., Pollak P.: How to enumerate the connectional isomers of a toridal polyhex fullerene. J. Chem. Inf. Comput. Sci. 38, 66–70 (1998) · doi:10.1021/ci970072i |
| [15] | Klein D.J.: Elemental benzenoids. J. Chem. Inf. Comput. Sci. 34, 453–459 (1994) · doi:10.1021/ci00018a037 |
| [16] | Klein D.J., Zhu H.: Resonance in elemental benzenoids. Discrete Appl. Math. 67, 157–173 (1996) · Zbl 0848.05061 · doi:10.1016/0166-218X(95)00017-L |
| [17] | Miškuf J., Jendroľ S.: On total edge irregularity strength of the grids. Tatra Mt. Math. Publ. 36, 147–151 (2007) |
| [18] | Nurdin, Salman, A.N.M., Baskoro, E.T.: The total edge-irregular strengths of the corona product of paths with some graphs. J. Combin. Math. Combin. Comput. 65, 163–175 (2008) · Zbl 1172.05033 |
| [19] | Ye, D., Qi, Z., Zhang, H.: On k-resonant fullerene graphs. SIAM J. Discrete Math. 23(2):1023–1044 (2009) · Zbl 1191.05090 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
