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ON THE DEGREE DISTANCE OF SOME COMPOSITE GRAPHS

Part of: Graph theory

Published online by Cambridge University Press:  04 October 2011

HONGBO HUA
HONGBO HUA*
Affiliation:
Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huaian, Jiangsu 223003, PR China Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China (email: [email protected])
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Abstract

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Let G be a connected simple graph. The degree distance of G is defined as D′(G)=∑ u∈V (G)dG(u)DG(u), where DG(u) is the sum of distances between the vertex u and all other vertices in G and dG(u) denotes the degree of vertex u in G. In contrast to many established results on extremal properties of degree distance, few results in the literature deal with the degree distance of composite graphs. Towards closing this gap, we study the degree distance of some composite graphs here. We present explicit formulas for D′ (G) of three composite graphs, namely, double graphs, extended double covers and edge copied graphs.

Keywords

degree distancedouble graphextended double coveredge copied graph

MSC classification

Secondary: 05C12: Distance in graphs 05C76: Graph operations (line graphs, products, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 85 , Issue 1 , February 2012 , pp. 164 - 171
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Alon, N., ‘Eigenvalues and expanders’, Combinatorica 6 (1986), 83–96.CrossRefGoogle Scholar
[2]Bondy, J. A. and Murty, U. S. R., Graph Theory with Applications (Macmillan, London, 1976).CrossRefGoogle Scholar
[3]Bucicovschia, O. and Cioabva, S. M., ‘The minimum degree distance of graphs of given order and size’, Discrete Appl. Math. 156 (2008), 3518–3521.CrossRefGoogle Scholar
[4]Chen, S. and Liu, W., ‘Extremal modified Schultz index of bicyclic graphs’, MATCH Commun. Math. Comput. Chem. 64 (2010), 767–782.Google Scholar
[5]Dankelmann, P., Gutman, I., Mukwembi, S. and Swart, H. C., ‘On the degree distance of a graph’, Discrete Appl. Math. 157 (2009), 2773–2777.CrossRefGoogle Scholar
[6]Dobrynin, A., Entringer, R. and Gutman, I., ‘Wiener index of trees: theory and applications’, Acta Appl. Math. 66 (2001), 211–249.Google Scholar
[7]Dobrynin, A. A. and Kochetova, A. A., ‘Degree distance of a graph: a degree analogue of the Wiener index’, J. Chem. Inf. Comput. Sci. 34 (1994), 1082–1086.Google Scholar
[8]Du, Z. and Zhou, B., ‘Degree distance of unicyclic graphs’, Filomat 24 (2010), 95–120.CrossRefGoogle Scholar
[9]Gutman, I., ‘Selected properties of the Schultz molecular topogical index’, J. Chem. Inf. Comput. Sci. 34 (1994), 1087–1089.CrossRefGoogle Scholar
[10]Horoldagva, B. and Gutman, I., ‘On some vertex-degree-based graph invariants’, MATCH Commun. Math. Comput. Chem. 65 (2011), 723–730.Google Scholar
[11]Ilić, A., Klavžar, S. and Stevanović, D., ‘Calculating the degree distance of partial Hamming graphs’, MATCH Commun. Math. Comput. Chem. 63 (2010), 411–424.Google Scholar
[12]Ilić, A., Stevanović, D., Feng, L., Yu, G. and Dankelmann, P., ‘Degree distance of unicyclic and bicyclic graphs’, Discrete Appl. Math. 159 (2011), 779–788.Google Scholar
[13]Khormali, O., Iranmanesh, A., Gutman, I. and Ahmadi, A., ‘Generalized Schultz index and its edge versions’, MATCH Commun. Math. Comput. Chem. 64 (2010), 783–798.Google Scholar
[14]Klavzǎr, S. and Gutman, I., ‘Wiener number of vertex-weighted graphs and a chemical application’, Discrete Appl. Math. 80 (1997), 73–81.CrossRefGoogle Scholar
[15]Tomescu, I., ‘Some extremal properties of the degree distance of a graph’, Discrete Appl. Math. 98 (1999), 159–163.CrossRefGoogle Scholar
[16]Tomescu, I., ‘Properties of connected graphs having minimum degree distance’, Discrete Math. 309 (2008), 2745–2748.Google Scholar
[17]Tomescu, I., ‘Ordering connected graphs having small degree distances’, Discrete Appl. Math. 158 (2010), 1714–1717.Google Scholar
[18]Tomescu, A. I., ‘Unicyclic and bicyclic graphs having minimum degree distance’, Discrete Appl. Math. 156 (2008), 125–130.Google Scholar
[19]Tomescu, A. I., ‘Minimal graphs with respect to the degree distance’, Technical Report, University of Bucharest, 2008, available online at: http://sole.dimi.uniud.it/∼alexandru.tomescu/files/dd-distance.pdf.Google Scholar
[20]Yan, W. and Yeh, Y.-N., ‘On the number of matchings of graphs formed by a graph operation’, Sci. China Ser. A 49 (2006), 1383–1391.Google Scholar