Some graph parameters of Indu-Bala product of graphs

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics, St Aloysius College, Edathua-689573,Kerala, India,

2 1Department of Mathematics, St. Stephens College, Uzhavoor - 686634, Kerala, India

3 Department of Mathematics, Bishop Chulaparambil Memorial(BCM) College, Kottayam - 686001, Kerala, India

4 Department of Mathematics, Marthoma College, Thiruvalla, 689 103, Kerala, India,

Abstract

The Indu-Bala product of graphs G and H consists of two disjoint copies of the join of G and H such that there is an  adjacency between the corresponding vertices in the two copies of H. A vertex subset S of a graph G = (V, E) is said to be a geodetic set if every vertex in G is in some u−v geodesic, where u and v are any two vertices in S. The minimum cardinality of such a set is the geodetic number of G. The vertex subset D of a graph G is said to be a dominating set if every vertex in G is either in D or adjacent to at least one vertex in D. The minimum cardinality of such a set is the domination number of G. In this work, the authors studied various geodetic and dominating extensions with respect to the Indu-Bala product of graphs. The Aα matrix associated with a graph is a convex linear combination of its adjacency matrix and degree diagonal matrix, offering deeper insights into the properties of both matrices. In this article the authors discuss the Aα spectrum of Indu-Bala product of graphs.

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References

[1] H. A. Ahangar and V. Samodivkin, The restrained geodetic number of a graph, Bull. Malaysian Math. Sci. Soc., 38 (2015) 1143–1155.
[2] D. Anandhababu and N. Parvathi, On independent domination number of Indu-Bala product of some families of graphs, AIP Conference Proceedings, 2112 (2019) 020139.
[3] R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, (Springer Science & Business Media, 2012).
[4] G. Chartrand, F. Harary, H. C. Swart, and P. Zhang, Geodomination in graphs, Bulletin of the ICA,  31 (2001), 51–59.
[5] G. Chartrand, F. Harary, and P. Zhang, On the geodetic number of a graph, Networks: An International Journal, 39 (2002), 1–6.
[6] M. Chellali, N. Jafari Rad, S. M. Sheikholeslami, and L. Volkmann, Varieties of Roman domination II, AKCE International Journal of Graphs and Combinatorics, 17 (03) (2020), 966–984.
[7] S. R. Chellathurai and S. P. Vijaya, Geodetic domination in the corona and join of graphs, Journal of Discrete Mathematical Sciences and Cryptography, 17(01) (2014), 81–90.
[8] E. J. Cockayne, R. Dawes, and S. T. Hedetniemi, Total domination in graphs, Networks , 10 (03) (1980), 211–219.
[9] E. J. Cockayne, P. A. Dreyer Jr, S. M. Hedetniemi, and S. T. Hedetniemi, Roman domination in graphs, Discrete mathematics, 278(03) (2004), 11–22.
[10] S. Y. Cui and G. X. Tian, The spectrum and the signless Laplacian spectrum of coronae, Linear Algebra Appl., 437 (2012), 1692–1703.
[11] P. J. Davis, Circulant Matrices, (John Wiley & Sons, 1979). 
[12] G. S. Domke, J. H. Hattingh, S. T. Hedetniemi, R. C. Laskar, and L. R. Markus, Restrained domination in graphs, Discrete mathematics, 203(03) (1999), 61–69.
[13] D.-Z. Du and P.-J. Wan, Connected Dominating Set: Theory and Applications, (Springer Science & Business Media, 2012).
[14] C. E. Go and S. R. Canoy Jr, Domination in the corona and join of graphs, International Mathematical Forum, 6 (2011), 763–771.
[15] C. D. Godsil and B. D. McKay, Constructing cospectral graphs, Aequationes Mathematicae, 25 (1982), 257–268.
[16] A. Hansberg and L. Volkmann, On the geodetic and geodetic domination numbers of a graph, Discrete Math., 310(15) (2010), 2140–2146.
[17] F. Harary, E. Loukakis, and C. Tsouros, The geodetic number of a graph, Math. Comput. Model., 17(11) (1993), 89–95.
[18] T. W. Haynes, S. T. Hedetniemi, and M. A. Henning, Topics in Domination in Graphs, (Springer, 2020).
[19] T. W. Haynes, S. T. Hedetniemi, and P. Slater, Fundamentals of Domination in Graphs, (CRC Press, 2013).
[20] G. Indulal and R. Balakrishnan, Distance spectrum of Indu–Bala product of graphs, AKCE International Journal of Graphs and Combinatorics, 13(3) (2016), 230–234.
[21] D.-X. Ma, X.-G. Chen, and L. Sun, On total restrained domination in graphs, Czechoslovak Mathematical Journal, 55 (2005), 165–173.
[22] V. Nikiforov, G. Past’n, O. Rojo, and R. L. Soto, On the Aα-spectra of trees, Linear Algebra and its Applications, 520 (2017), 286–305.
[23] S. Patil and M. Mathapati, Spectra of Indu–Bala product of graphs and some new pairs of cospectral graphs, Discrete Mathematics, Algorithms and Applications, 11(5) (2019), 1950056.
[24] I. M. Pelayo, Geodesic Convexity in Graphs, (Springer, 2013). 
[25] M. Priyadharshini, N. Parvathi, and I. N. Cangul, Independent strong domination number of Indu-Bala product of graphs, Palestine Journal of Mathematics, 12(02) (2023), 80–88.
[26] D. B. West et al., Introduction to Graph Theory, (Prentice hall Upper Saddle River, 2001). 
[27] F. Z. Zhang, The Schur Complement and its Applications, (Springer, 2005).
[28] R.W.Hung, Restrained domination and its variants in extended supergrid graphs, Th. Comp. Sci, (2023), 113-132.
[29] K.L.Bhavyavenu, Some study on geodetic number of a graph, Ph.D Thesis, http://hdl.handle.net/10603/363058, (2020)
Journal of Hyperstructures

Articles in Press, Accepted Manuscript
Available Online from 28 April 2025

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