Document Type : Original Article

**Author**

Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785 – 136, I R. Iran

**Abstract**

The eccentric distance sum is a graph invariant defined as $\sum_{uv\in E} ε_{G}(v)D_{G}(v)$, where ε_{G}(v) is the eccentricity of a vertex v in G and D_{G}(v ) is the sum of distances of all vertices in G from v. In this paper, we compute the eccentric distance sum of Volkmann tree and then we obtain some results for vertex−transitive graphs

**Graphical Abstract**

**Keywords**

[1]. A. R. Ashra i, M. Saheli and M. Ghorbani, The eccentric connectivity index of nanotubes and nanotori, Journal of Computational and Applied Mathematics, 235 (2011), 4561 – 4566.

[2] T. Doslić, M. Ghorbani and M. Hosseinzadeh, Eccentric connectivity polynomial of some graph operations, Utilitas Mathematica, 84 (2011), 297-309.

[3] M. Ghorbani, Connective eccentric index of fullerenes, J. Math. NanoSci., 1 (2011), 43–52.

[4] M. Songhori, ccentric connectivity index of fullerene graphs, J. Math. NanoSci., 2 (2011), 21–27.

[5] V. Sharma, R. Goswami and A. K. Madan, Eccentric connectivity index: Anovel highly discriminating topological descriptor for structure-property andstructureactivity studies, J. Chem. Inf. Model., 37 (1997), 273 – 282.

[6] S. Gupta, M. Singh and A.K. Madan, Eccentric distance sum: a novel graph invariant for predicting biological and physical properties, J. Math. Anal. Appl., 275 (2002), 386 – 401.

[7] A. Ilić, G. Yu and L. Feng, On the eccentric distance sum of graphs, J. Math. Anal. Appl., 381 (2011), 590 – 600.

[8] B. E. Sagan, Y.-N. Yeh and P. Zhang, The Wiener polynomial of a graph, Int. J. Quantum. Chem., 60 (1996), 959 – 969.

[9] D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996.

June 2012

Pages 37-41

**Receive Date:**04 December 2011**Revise Date:**21 February 2012**Accept Date:**11 May 2012**First Publish Date:**01 June 2012