Zagreb Radio Indices of Some Graphs with Diameter Two

A.T. Raj¹ , J. V. Kureethara²

1. Dept. Mathematics, St. Teresa’s College, Ernakulam, Kerala, India.
2. Dept. Mathematics and Statistics, Christ University, Bengaluru, Karnataka, India.

Correspondence should be addressed to: frjoseph@christuniversity.in .

Section:Research Paper, Product Type: Isroset-Journal
Vol.4 , Issue.6 , pp.7-11, Dec-2017

CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v4i6.711

Online published on Dec 31, 2017

Copyright © A.T. Raj, J. V. Kureethara . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
A good amount of studies are there to explore the possibilities of non-laboratory characterization of molecules. In the field of quantitative structure-activity relationships (QSAR), topological indices play an important role. This paper is a perusal of the recently introduced topological indices called Zagreb radio indices. The Zagreb radio indices make use of the labeling of the vertices from the multi-level distance labeling of vertices called radio coloring. This is a vertex labeling with the condition on the distance between the vertices. Radio coloring is a non-negative integer labeling of the vertices of a graph such that difference of the colors of any pair of vertices is at least the difference between the diameter of the graph and the distance between those vertices plus one. This condition forces every vertex to have distinct labels. If the diameter of the graph is one, then radio coloring is nothing but the proper coloring. The Zagreb indices were originally introduced for the degrees of the vertices. Here, the Zagreb radio indices of some graphs of diameter two are computed replacing the degrees of the vertices with radio labels of the vertices originated from the minimal radio coloring.

Key-Words / Index Term :
Zagreb Indices, Zagreb Radio Indices, Distance Labeling, Mathematical Chemistry, Diameter 2 graphs

References :
[1] D. B. West, "Introduction to Graph Theory", 2 ed., New Delhi: PHI Learning Private Ltd, 2012.
[2] H. Hosaya, "Topological Index. A Newly Proposed Quantity Characterizing the Topological Nature of Structural Isomers of Saturated Hydrocarbons," "Bulletin of the Chemical Society of Japan", vol. 44, no. 9, pp. 2332-2339, 1971.
[3] I. Gutman, "Degree-based topological indices," "Croatica Chemica Acta", vol. 86, no. 4, pp. 351-361, 2013.
[4] I. Gutman and N. Trinajstić, "Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons," "Chemical Physics Letters", vol. 17, no. 4, pp. 535-538, 1972.
[5] A. T. Balaban, I. Motoc, D. Bonchev and O. Mekenyan, "Topological indices for structure-activity correlations," in "Steric Effects in Drug Design", Berlin, Springer-Verlag, pp. 21-55, 1983.
[6] A. T. Raj and J. V. Kureethara, "Zagreb Radio Indices of Graphs," unpublished, 2017.
[7] G. Chartrand, D. Erwin, F. Harary and P. Zhang, "Radio labelings of graphs," "Bulletin of the Institute of Combinatorics and its Applications", vol. 33, pp. 77-85, 2001.
[8] G. Chartrand and P. Zhang, "Colorings, Distance and Domination," in "Chromatic Graph Theory", Florida, CRC Press, p. 415, 2009.
[9] R. K. Yeh, "Labeling graphs with a condition at distance two", Columbia, SC: Department of Mathematics, University of South Carolina, 1990.