Integrity of Generalized Transformation Graphs

Bommanahal Basavanagoud, Shruti Policepatil


The values of vulnerability helps the network designers to construct such a communication network which remains stable after some of its nodes or communication links are damaged. The transformation graphs considered in this paper are taken as model of the network system and it reveals that, how network can be made more stable and strong. For this purpose the new nodes are inserted in the network. This construction of new network is done by using the definition of generalized transformation graphs of a graphs. Integrity is one of the best vulnerability parameter.
In this paper, we investigate the integrity of generalized transformation graphs and their complements. Also, we find integrity of semitotal point graph of combinations of basic graphs. Finally, we characterize few graphs having equal integrity values as that of generalized transformation graphs of same structured graphs.


Vulnerability; connectivity; integrity; generalized transformation graphs; semitotal point graph

Full Text:



M. Atici, R. Crawford, The integrity of small cage graphs, Australas. J. Combin., 43, 2009, 39--55.

M. Atici, A. Kirlangic, Counter examples to the theorems of integrity of prism and ladders, J. Combin. Math. Combin. Comp., 34, 2000, 119--127.

M. Atici, R. Crawford, C. Ernst, New upper bounds for the integrity of cubic graphs, Int. J. Comput. Math., 81(11), 2004, 1341--1348.

A. Aytac, S. Celik, Vulnerability: Integrity of a middle graph, Selcuk J. Appl. Math., 9(1), 2008, 49--60.

K. S. Bagga, L. W. Beineke, W. D. Goddard, M. J. Lipman, R. E. Pippert, A survey of integrity, Discrete Appl. Math. 37, 1992, 13--28.

C. A. Barefoot, R. Entringer, H. C. Swart, Vulnerability in graphs - A comparitive survey, J. Combin. Math. Combin. Comput., 1, 1987, 13--21.

C. A. Barefoot, R. Entringer, H. C. Swart, Integrity of trees and powers of cycles, Congr. Numer., 58, 1987, 103--114.

B. Basavanagoud, I. Gutman, C. S. Gali, Second Zagreb index and coindex of some derived graphs, Kragujevac J. Sci., 37, 2015, 113--121.

B. Basavanagoud, I. Gutman, V. R. Desai, Zagreb indices of generalized transformation graphs and their complements, Kragujevac J. Sci., 37, 2015, 99--112.

B. Basavanagoud, S. M. Hosamani, S. H. Malghan, Domination in semitotal point graph, J. Comp. Math. Sci., 1(5), 2010, 598--605.

P. Dundar, A. Aytac, Integrity of total graphs via certain parameters, Math. Notes, 76(5), 2004, 665--672.

W. Goddard, On the Vulnerability of Graphs, Ph.D. Thesis, University of Natal, Durban, S.A., 1989.

W. D. Goddard, H. C. Swart, On the integrity of combinations of graphs, J. Combin. Math. Combin. Comp., 4, 1988, 3--18.

W. Goddard, H. C. Swart, Integrity in graphs: Bounds and basics, J. Combin. Math. Combin. Comp., 7, 1990, 139--151.

F. Harary, Graph Theory, Addison-Wesely, Reading Mass, 1969.

V. R. Kulli, College Graph Theory, Vishwa Int. Publ., Gulbarga, India, 2012.

H. S. Ramane, B. Basavanagoud, R. B. Jummannavar, Harmonic index and Randic index of generalized transformation graphs, J. Nigerian Math. Soc., 37(2), 2018, 57--69.

E. Sampathkumar, S. B. Chikkodimath, Semitotal graphs of a graph, J. Karnatak Univ. Sci., 18, 1973, 274--280.

A. Vince, The integrity of a cubic graph, Discrete Appl. Math., 140, 2004, 223--239.



  • There are currently no refbacks.

Copyright (c) 2021 Shruti Policepatil, B. Basavanagoud

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Ratio Mathematica - Journal of Mathematics, Statistics, and Applications. ISSN 1592-7415; e-ISSN 2282-8214.