DEGREE-BASED INVARIANTS OF MYCIELSKI CONSTRUCTION: IRREGULARITY, TOTAL IRREGULARITY, VARIANCE

Zeynep Nihan Berberler

DEGREE-BASED INVARIANTS OF MYCIELSKI CONSTRUCTION: IRREGULARITY, TOTAL IRREGULARITY, VARIANCE

Abstract

The degree-based graph invariants are parameters defined by degrees of vertices. A graph is regular if all of its vertices have the same degree. Otherwise a graph is irregular. To measure how irregular a graph is, graph topological indices were proposed including the irregularity of a graph, total irregularity of a graph, and the variance of the vertex degrees. In this paper, the above mentioned irregularity measures for Mycielski constructions of any underlying graph are considered and exact formulae are derived.

Keywords

References

[1] Albertson M.O., (1997) The irregularity of a graph, Ars Combin. 46, 219–225.
[2] Abdo H., Brandt S., Dimitrov D., (2014) The total irregularity of a graph, Discrete Math. Theor. Comput. Sci. 16(1), 201-206.
[3] Bell F. K., (1992) A note on the irregularity of graphs, Linear Algebra Appl. 161, 45–54.
[4] Berberler Z. N., (2019) On computing the irregularity of R-corona product types of graphs, Util. Math. to appear.
[5] Berberler Z. N., (2019) Total Irregularity of Indu-Bala Product of Graphs, APJES 7(1), 52-55.
[6] Criado R., Flores J., García del Amo A., Romance M., (2014) Centralities of a network and its line graph: an analytical comparison by means of their irregularity, Int. J. Comput. Math. 91(2), 304-314.
[7] Estrada E., (2010) Randić index, irregularity and complex biomolecular networks, Acta Chim. Slov. 57, 597–603.
[8] Hansen P., Mélot H., (2005) Variable neighborhood search for extremal graphs. 9. Bounding the irregularity of a graph, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 69, 253-264.

[9] You L.H., Yang J.S., You Z.F., (2014) The maximal total irregularity of unicyclic graphs, Ars Comb. 114, 153–160.
[10] You L.H., Yang J.S., Zhu Y.X., You Z.F., (2014) The maximal total irregularity of bicyclic graphs, J. Appl. Math. Article ID 785084, http://dx.doi.org/10.1155/2014/785084.
[11] Zhu Y., You L., Yang J., (2016) The minimal total irregularity of some classes of graphs, Filomat 30(5), 1203-1211.
[12] Dimitrov D., Škrekovski R., (2015) Comparing the irregularity and the total irregularity of graphs, Ars Math. Contemp. 9, 45–50.
[13] Zhou B., (2008) On irregularity of graphs, Ars Combin. 88, 55-64.
[14] Abdo H., Dimitrov D., (2014) The irregularity of graphs under graph operations, Discuss. Math. Graph Theo. 34(2), 263–278.
[15] Abdo H., Dimitrov D., (2014) The total irregularity of graphs under graph operations, Miskolc Math. Notes 15, 3–17.
[16] Abdo H., Dimitrov D., (2015) The Total Irregularity of Some Composite Graphs, Int. J. Comput. Appl. 122(21), 1-9.
[17] Abdo H., Dimitrov D., Gao W., (2017) On the irregularity of some molecular structures, Can. J. Chem. 95(2), 174-183.
[18] Abdo H., Cohen N., Dimitrov D., (2014) Graphs with maximal irregularity, Filomat 28(7), 1315-1322.
[19] Gutman I., Hansen P., Mélot H., (2005) Variable neighborhood search for extremal graphs. 10. Comparison of irregularity indices for chemical trees, J. Chem. Inf. Model. 45, 222-230.
[20] Henning M.A., Rautenbach D., (2007) On the irregularity of bipartite graphs, Discrete Math. 307, 1467-1472.
[21] Albertson M.O., Berman D., (1991) Ramsey graphs without repeated degrees, Congr. Numer. 83, 91–96.
[22] Tavakoli M., Rahbarnia F., Ashrafi A.R., (2014) Some new results on irregularity of graphs, J. Appl. Math. Inform. 32, 675-685.
[23] Luo W., Zhou B., (2010) On the irregularity of trees and unicyclic graphs with given matching number, Util. Math. 83, 141-147.
[24] Mycielski J., (1955) Sur le coloriage des graphs, Colloq. Math. 3, 161–162.
[25] Gutman I., Trinajstić N., (1972) Graph theory and molecular orbitals. Total  - electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17, 535-538.

Details

Primary Language

English

Subjects

Engineering

Journal Section

Research Article

Authors

Zeynep Nihan Berberler This is me
0000-0001-9179-3648
Türkiye

Publication Date

September 1, 2020

Submission Date

March 20, 2018

Acceptance Date

March 5, 2019

Published in Issue

Year 2019 Volume: 37 Number: 3

IZ

https://izlik.org/JA23KR48MP

Cite

RIS / Bibtex

APA

Berberler, Z. N. (2020). DEGREE-BASED INVARIANTS OF MYCIELSKI CONSTRUCTION: IRREGULARITY, TOTAL IRREGULARITY, VARIANCE. Sigma Journal of Engineering and Natural Sciences, 37(3), 747-754. https://izlik.org/JA23KR48MP

AMA

1.Berberler ZN. DEGREE-BASED INVARIANTS OF MYCIELSKI CONSTRUCTION: IRREGULARITY, TOTAL IRREGULARITY, VARIANCE. SIGMA. 2020;37(3):747-754. https://izlik.org/JA23KR48MP

Chicago

Berberler, Zeynep Nihan. 2020. “DEGREE-BASED INVARIANTS OF MYCIELSKI CONSTRUCTION: IRREGULARITY, TOTAL IRREGULARITY, VARIANCE”. Sigma Journal of Engineering and Natural Sciences 37 (3): 747-54. https://izlik.org/JA23KR48MP.

EndNote

Berberler ZN (September 1, 2020) DEGREE-BASED INVARIANTS OF MYCIELSKI CONSTRUCTION: IRREGULARITY, TOTAL IRREGULARITY, VARIANCE. Sigma Journal of Engineering and Natural Sciences 37 3 747–754.

IEEE

[1]Z. N. Berberler, “DEGREE-BASED INVARIANTS OF MYCIELSKI CONSTRUCTION: IRREGULARITY, TOTAL IRREGULARITY, VARIANCE”, SIGMA, vol. 37, no. 3, pp. 747–754, Sept. 2020, [Online]. Available: https://izlik.org/JA23KR48MP

ISNAD

Berberler, Zeynep Nihan. “DEGREE-BASED INVARIANTS OF MYCIELSKI CONSTRUCTION: IRREGULARITY, TOTAL IRREGULARITY, VARIANCE”. Sigma Journal of Engineering and Natural Sciences 37/3 (September 1, 2020): 747-754. https://izlik.org/JA23KR48MP.

JAMA

1.Berberler ZN. DEGREE-BASED INVARIANTS OF MYCIELSKI CONSTRUCTION: IRREGULARITY, TOTAL IRREGULARITY, VARIANCE. SIGMA. 2020;37:747–754.

MLA

Berberler, Zeynep Nihan. “DEGREE-BASED INVARIANTS OF MYCIELSKI CONSTRUCTION: IRREGULARITY, TOTAL IRREGULARITY, VARIANCE”. Sigma Journal of Engineering and Natural Sciences, vol. 37, no. 3, Sept. 2020, pp. 747-54, https://izlik.org/JA23KR48MP.

Vancouver

1.Zeynep Nihan Berberler. DEGREE-BASED INVARIANTS OF MYCIELSKI CONSTRUCTION: IRREGULARITY, TOTAL IRREGULARITY, VARIANCE. SIGMA [Internet]. 2020 Sep. 1;37(3):747-54. Available from: https://izlik.org/JA23KR48MP