# zbMATH — the first resource for mathematics

Hyper- and reverse-Wiener indices of F-sums of graphs. (English) Zbl 1221.05120
Summary: The Wiener index $$W(G)=\sum _{\{u,v\} \subset V(G)}d(u,v)$$, the hyper-Wiener index $$WW(G) = \frac{1}{2} \sum _{\{u,v\} \subset V(G)} [d(u, v) + d^2(u, v)]$$ and the reverse-Wiener index $$\Lambda (G) = \frac{n(n-1)D}{2} - W(G)$$, where $$d(u,v)$$ is the distance of two vertices $$u,v$$ in $$G, d^{2}(u,v)=d(u,v)^{2}, n=|V(G)|$$ and $$D$$ is the diameter of $$G$$.
M. Eliasi and B. Taeri [“Four new sums of graphs and their Wiener indices,” Discrete Appl. Math. 157, No. 4, 794–803 (2009; Zbl 1172.05318)] introduced the F-sums of two connected graphs. In this paper, we determine the hyper- and reverse-Wiener indices of the F-sum graphs and, subject to some condition, we present some exact expressions of the reverse-Wiener indices of the F-sum graphs.

##### MSC:
 05C12 Distance in graphs 05C76 Graph operations (line graphs, products, etc.)
Full Text:
##### References:
 [1] Balaban, A.T.; Mills, D.; Ivanciuc, O.; Basak, S.C., Reverse Wiener indices, Croat. chem. acta, 73, 923-941, (2000) [2] Cai, X.; Zhou, B., Reverse Wiener indices of connected graphs, MATCH commun. math. comput. chem., 60, 95-105, (2008) · Zbl 1246.05045 [3] Cash, G.G., Relationship between the hosoya polynomial and the hyper-Wiener index, Appl. math. lett., 15, 893-895, (2002) · Zbl 1009.05127 [4] Cash, G.G., Polynomial expressions for the hyper-Wiener index of extended hydrocarbon networks, Comput. chem., 25, 577-582, (2001) [5] Cvetkocić, D.M.; Doob, M.; Sachs, H., Spectra of graphs—theory and application, (1980), Academic Press NewYork [6] Diudea, M.V.; Gutman, I.; Jantschi, L., Molecular topology, (2001), Huntington NY [7] Dobrynin, A.A.; Entringer, R.; Gutman, I., Wiener index of trees: theory and applications, Acta appl. math., 66, 211-249, (2001) · Zbl 0982.05044 [8] Dobrynin, A.A.; Gutman, I.; Klavzar, S.; Zigert, P., Wiener index of hexagonal systems, Acta appl. math., 72, 247-294, (2002) · Zbl 0993.05059 [9] Eliasi, M.; Taeri, B., Four new sums of graphs and their Wiener indices, Discrete appl. math., 157, 794-803, (2009) · Zbl 1172.05318 [10] Gutman, I., Relation between hyper-Wiener and Wiener index, Chem. phys. lett., 364, 352-356, (2002) [11] Gutman, I.; Dobrynin, A.A.; Klavžar, S.; Pavlović, L., Wiener-type invariants of trees and their relation, Bull. inst. combin. appl., 40, 23-30, (2004) · Zbl 1036.05020 [12] Gutman, I.; Žerovnik, J., Corroborating a modification of the Wiener index, Croat. chem. acta, 75, 603-612, (2002) [13] Imrich, W.; Klavzar, S., Product graphs: structure and recognition, (2000), John Wiley & Sons New York, USA [14] Ivanciuc, O.; Ivanciuc, T.; Balaban, A.T., Quantitative structure-property relationshipe valuation of structural descriptors derived from the distance and reverse Wiener matrices, Internet electron. J. mol. des., 1, 467-487, (2002) [15] Ivanciuc, O.; Ivanciuc, T.; Klein, D.J.; Seitz, W.A.; Balaban, A.T., Wiener index extension by counting even/odd graph distances, J. chem. inf. comput. sci., 41, 536-549, (2001) [16] Klavzar, S.; Gutman, I., A theorem on Wiener-type invariants for isometric subgraphs of hypercubes, Appl. math. lett., 19, 1129-1133, (2006) · Zbl 1120.05026 [17] Klavzar, S.; Zigert, P.; Gutman, I., Analgorithm for the calculation of the hyper-Wiener index of benzenoid hydrocarbons, Comput. chem., 24, 229-233, (2000) · Zbl 1034.92040 [18] Klein, D.J.; Lukovits, I.; Gutman, I., On the definition of the hyper-Wiener index for cycle-containing structures, J. chem. inf. comput. sci., 35, 50-52, (1995) [19] Li, X.; Jalbout, A.F., Bond order weighted hyper-Wiener index, J. mol. structure (theochem), 634, 121-125, (2003) [20] Luo, W.; Zhou, B., Further properties of reverse Wiener index, MATCH commun. math. comput. chem., 61, 653-661, (2009) · Zbl 1224.05149 [21] Nikolic, S.; Trinajstić, N.; Randić, M., Wiener index revisited, Chem. phys. lett., 333, 319-321, (2001) [22] Trinajstic, N., Chemical graph theory, (1992), CRC Press Boca Raton, FL [23] Vukicević, D.; Žerovnik, J., Variable Wiener indices, MATCH commun. math. comput. chem., 53, 385-402, (2005) · Zbl 1088.05029 [24] Wiener, H., Structural determination of the paraffin boiling points, J. amer. chem. soc., 69, 17-20, (1947) [25] Zhang, B.; Zhou, B., Modified and reverse Wiener indices of trees, Z. nat.forsch., 61a, 536-540, (2006) [26] Zhou, B.; Gutman, I., Relations between Wiener, hyper-Wiener and Zagreb indices, Chem. phys. lett., 394, 93-95, (2004)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.