On Domination Zagreb Polynomials of Graphs and
Some Graph Operations

Ashwini A. S; Hanan Ahmed; Soner Nandappa D

doi:https://doi.org/10.14445/22315373/IJMTT-V68I2P511

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 68 | Issue 2 | Year 2022 | Article Id. IJMTT-V68I2P511 | DOI : https://doi.org/10.14445/22315373/IJMTT-V68I2P511

On Domination Zagreb Polynomials of Graphs and Some Graph Operations

Ashwini A. S, Hanan Ahmed, Soner Nandappa D

Abstract

Graph polynomials are powerful and well-developed tools to express graph parameters. A set D ⊂ V (G) is a dominating set of G if every vertex not in D is adjacent to at least one vertex in D. The minimal domination set is the dominating set such that D − v is not dominating set. H. Ahmed et al. [4] invented the domination Zagreb Polynomials as follows: 𝐷𝑀1 (𝐺, 𝑥) = ∑ 𝑥 𝑑𝑑𝐺 2(𝑢) 𝑢𝜖𝑉(𝐺) , 𝐷𝑀2 (𝐺, 𝑥) = ∑ 𝑥 𝑑𝑑𝐺 (𝑢)𝑑𝑑𝐺 (𝑣) 𝑢𝑣𝜖𝐸(𝐺) , 𝐷𝑀1 ∗ (𝐺, 𝑥) = ∑ 𝑥 𝑑𝑑𝐺 (𝑢)+𝑑𝑑𝐺 (𝑣) 𝑢𝑣𝜖𝐸(𝐺) . Where dd(v) is the domination degree of a vertex v ∈ V(G) which is defined as the number of minimal dominating sets of G contains v. In this paper, we calculate the domination Zagreb polynomials for some important families of graphs and some graph operations like corona product and join of graphs. Some properties of the domination Zagreb polynomials of graphs are also established.

Keywords

Domination degrees, Domination Zagreb polynomial, Graph operation

References

[1] H. Ahmed, A. Alwardi, R. M. Salestina and N. D. Soner, Domination, ᵧ - Domination Topological Indices and Φp -Polynomial of Some Chemical Structures Applied for The Treatment of Covid-19 Patients, Biointerface Research in Applied Chemistry, 11 (2021) 13290-13302. Doi: 10.33263/Briac115.1329013302.
[2] H. Ahmed, A. Alwardi, S.Wazzan, Domination Topological Properties of Polyhydroxybutyrate and Polycaprolactone with Qsrp Analysis, Nanosystem: Physics Chemistry Mathematics, 12(6) (2021) 664-671. Doi:10.17536/2220-8054-2021-12-6-664-671.
[3] H. Ahmed, M. R. Farahani, A. Alwardi, R. M. Salestina, Domination Topological Properties of Some Chemical Structures Using Φp -Polynomial Approach , Euresian Chemical Communications, 3 (2021) 210-218. Doi: 10.22034/Ecc.2021.271992.1133.
[4] H. Ahmed, R. M. Salestina And A. Alwardi, Domination Topological Indices and Their Polynomials of A Firefly Graph, Journal of Discrete Mathematical Sciences and Cryptography, 24(2) (2021) 325-341. Doi: 10.1080/09720529.2021.1882155.
[5] H. Ahmed, R. M. Salestina, A. Alwardi, N. D. Soner, Forgotten Domination, Hyper Domination and Modified Forgotten Domination Indices of Graph, Journal of Discrete Mathematics Sciences and Cryptography, 24(2) (2021) 353-368. Doi:10.1080/09720529.2021.1885805
[6] A. Alameri, Second Hyper-Zagreb Index of Titania Nanotubes and Their Applications, Ieee Access, (2021) 9567- 9571. Doi: 10.1109/Access.2021.3050774
[7] A. Alsinai, H. Ahmed, A. Alwardi, & N. D. Soner. (2021). Hdr Degree Based Indices and Mhr-Polynomial for The Treatment of Covid-19, Biointerface Research In Applied Chemistry, 6(12) (2022) 7214 – 7225. Available: Https://Doi.Org/10.33263/Briac126.72147225
[8] A. Alsinai, A. Alwardi, & N. D. Soner. (2021). On The Ψk-Polynomial of Graph. Eurasian Chem. Commun, 3 219-226. Available: Https://Doi.Org/10.22034/Ecc.2021.274069.1138
[9] A. Ayache, A. Alameri, Molecular Topological Properties of Spider's Web Graph and Their Possible Applications, Commun. Combin., Cryptogr. & Computer Sci., 1 (2021) 17–26.
[10] A. Ayache, A. Alameri, M. Alsharafi, H. Ahmed , The Second Hyper-Zagreb Coindex of Chemical Graphs and Some Applications, Journal of Chemistry, 2021(2) 8. Doi: 10.1155/2021/3687533
[11] A. Ayache, A. Alameri, A. S. Ghallab, A. Modabish , Wiener Polynomial And Wiener Index of Conical Graphs, Sylwan, 11(3) (2020) 107-116.
[12] S. C. Basak, V. R. Magnuson, G. J. Niemi, R. R. Regal and G. D. Veith, Topological Indices; Their Nature, Mutual Relatedness And Applications, International Journal of Mathematical Modelling, 8 (1987) 300-305. Doi: 10.1016/0270-0255(87)90594-X.
[13] B. Basavanagouda And E. Chaitra, Zagreb Polynomials of Graph Operations, International Journal of Applied Engineering Research, 15(3) (2020) 287-293.
[14] B. Borovi_Canin, K. C. Das, B. Furtual And I. Gutman, Bounds For Zagreb Indices, Match Commun. Math. Comput. Chem., 78 (2017) 17-100.
[15] M. Faisal Nadeem, Sohail Zafar, Zohaib Zahid, On Certain Topological Indices of The Line Graph of Subdivision Graph, Appl. Math. Comput., 457(2015) 790-794. Doi:10.1.06/J.Amc.2015.09.016.
[16] I. Gutman, Kragujevac Trees and Their Energy, Appl. Math. Inform. Mech., 6(2) (2014) 71-79. Doi: 10.5937/Spsunp1402071g.
[17] A. M. Hanan Ahmed, A. Alwardi, R. M. Salestina, On Domination Topological Indices of Graphs, International Journal Of Analysis And Applocations, 19(1) (2021) 47-64. Doi: 10.28924/2291-8639-19-2021-47.
[18] F. Harary, Graphs Theory, Addison-Wisley Publishing Co., Reading Mass, (1969).
[19] T. W. Haynes , S. T. Hedetniemi and P. J. Slater, Domination In Graphs: Advanced Topics, New York: Marcel Dekkar, Inc. (1998).
[20] S. M. Hedetniemi, S. T. Hedetniemi, R. Laskar, A. A. Mcrae And C. Wallis, Domination Partition of Graphs, J. Combin, Inform, Systems Sci., 34(1-4) (2009) 183-192
[21] S. Javarajua, H. Ahmed, A. Alsinai, N. D. Soner, Domination Topological Properties of Carbidopa-Levodopa Used for Treatment Parkinson’s Disease By Using Φp-Polynomial.Eurasian Chem. Commun, 3(9) (2021) 614-621. Available: Https://Doi.Org/10.22034/Ecc.2021.295039.1203
[22] S. Javarajua, A. Alsinai, A. Alwardi, H. Ahmed, N. D. Soner, Reciprocal Leap Indices of Some Wheel Related Graphs, Journal of Prime Research on Mathematics, 17(2) (2021) 100-110.
[23] V. R. Kulli, New Irregularity Sombor Indices And New Adriatic (A, B)-Ka Indices Of Certain Chemical Drugs, International Journal of Mathematics Trends And Technology, 67(9) (2021) 105-113. Doi: 10.14445/22315373/Ijmtt-V67|9p512
[24] V. R. Kulli, Different Versions of Nirmala Index of Certain Chemical Structures, International Journal of Mathematics Trends and Technology, 67(7) (2021) 56-63. Doi: 10.14445/22315373/Ijmtt-V67i7p507
[25] I. Rusu, J. Spinrad, Domination Graphs: Examples and Counterexamples, Discrete Applied Mathematics, 110(2-3) (2001) 289-300. Doi: 10.1016/S0166-218x(00)00274-2.
[26] V. S. Thakur, S. Shobha, Domination And It's Type in Graphs Theory, Jetir, 7(3) (2020) 1549-1557.

Citation :

Ashwini A. S, Hanan Ahmed, Soner Nandappa D, "On Domination Zagreb Polynomials of Graphs and Some Graph Operations," International Journal of Mathematics Trends and Technology (IJMTT), vol. 68, no. 2, pp. 66-74, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I2P511