The status σ (*u*) of a vertex *u* is defined as the sum of the distances between *u* and all other vertices of a graph *G*. In this paper we have defined the harmonic status index of a graph *G* as and obtained the bounds for *HS*(*G*). Further the harmonic status index of some graphs is obtained.

Periodical:

Bulletin of Mathematical Sciences and Applications (Volume 17)

Pages:

24-32

Citation:

H. S. Ramane et al., "Harmonic Status Index of Graphs", Bulletin of Mathematical Sciences and Applications, Vol. 17, pp. 24-32, 2016

Online since:

November 2016

Keywords:

Distribution:

Open Access

This work is licensed under a

Creative Commons Attribution 4.0 International License

References:

[1] J.A. Bondy, U.S.R. Murty, Graph theory with applications. American Elsevier Publishing Co., New York, (1976).

[2] F. Buckley, F. Harary, Distance in Graphs, Addison–Wesley, Redwood, (1990).

[3] R. Chang, Y. Zhu, On the harmonic index and the minimum degree of a graph, Romanian J. Inf. Sci. Tech. 15 (2012) 335-343.

[4] K.C. Das, I. Gutman, Estimating the Wiener index by means of number of vertices of edges and diameter, MATCH Commun. Math. Comput. Chem. 64 (2010) 647-660.

[5] H. Deng et al., On the harmonic index and the chromatic number of a graph, Discrete Appl. Math. 161 (2013) 2740-2744.

[6] J. Devillers, A.T. Balaban (Eds. ), Topological Indices and Related Descriptors in QSAR and QSPR, Gordan and Breach, Amsterdam, (1999).

[7] A.A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: Theory and Applications, Acta Appl. Math. 66 (2001) 211-249.

[8] S. Fajtlowicz, On conjectures of Graffiti - II, Congr. Numer. 60 (1987) 187-197.

[9] I. Gutman, L. Pavlovic, The energy of some graphs with large number of edges. Bull. Acad. Serbe Sci. Arts. (Cl. Math. Natur. ) 118 (1999) 35-50.

[10] I. Gutman et al., Some recent results in the Theory of the Wiener number, Indian J. Chem. 32A (1993) 651-661.

[11] F. Harary, Status and Contrastatus, Sociometry. 22 (1959) 23-43.

[12] Y. Hu, X. Zhou, On the harmonic index of the unicyclic and bicyclic graphs, Wseas Tran. Math. 12 (2013) 716-726.

[13] G. Indulal, I. Gutman, A. Vijaykumar, On the distance energy of a graph, MATCH Commun. Math. Comput. Chem. 60 (2008) 461-472.

[14] O. Ivanciuc, T.S. Balaban, A.T. Balaban, Design of topological indices, Part 4. Reciprocal distance matrix related local vertex invariants and topological indices, J. Math. Chem. 12 (1993) 309-318.

[15] J. Liu, On the harmonic index of triangle free graphs, Appl. Math. 4 (2013) 1204-1206.

[16] J. Liu, On the harmonic index and diameter of graphs, J. Appl. Math. Phy. 1 (2013) 5-6.

[17] D. Plavsic, S. Nikolic, N. Trinajstic, On the Harary index for the characterization of chemical graphs, J. Math. Chem. 12 (1993) 235-250.

[18] H.S. Ramane, Distance energy of graphs, in: Energies of Graphs – Theory and Applications (Eds.: I. Gutman, X. Li), Univ. Kragujevac, Kragujevac, 2016, p.123 – 144.

[19] H.S. Ramane, A.B. Ganagi, H.B. Walikar, Wiener index of graphs in terms of eccentricities, Iranian J. Math. Chem. 4 (2013) 239-248.

[20] H.S. Ramane, V.V. Manjalapur, Note on the bounds on Wiener number of a graph, MATCH Commun. Math. Comput. Chem. 76 (2016) 19-22.

[21] M. Randic, Novel molecular descriptor for structure-property studies, Chem. Phys. Lett. 211 (1993) 478-483.

[22] B. Shwetha Shetty, V. Lokesha, P.S. Ranjini, On the harmonic index of graph operations, Trans. Combin. 4 (2015) 5-14.

[23] H.B. Walikar, V.S. Shigehalli, H.S. Ramane, Bounds on the Wiener number of a graph, MATCH Commun. Math. Comp. Chem. 50 (2004) 117-132.

[24] H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc. 69 (1947) 17-20.

[25] L. Zhong, The harmonic index of graphs, Appl. Math. Lett. 25 (2012) 561-566.

[26] Y. Zhu, R. Chang, On the harmonic index of bicyclic conjugated molecular graphs, Filomat. 28(2) (2014) 421-428.