Steiner radial number resulting from various graph operations
Abstract
The Steiner n-radial graph of a graph G on p vertices, denoted by SRn(G), has the vertex set as in G and any n(2 ≤ n ≤ p) vertices are mutually adjacent in SRn(G) if and only if they are n-radial in G. When G is disconnected, any n vertices are mutually adjacent in SRn(G) if not all of them are in the same component. For the edge set of SRn(G), draw Kn corresponding to each set of n-radial vertices. The Steiner radial number rS(G) of a graph G is the least positive integer n such that the Steiner n-radial graph of G is complete. In this paper, Steiner radial number has been determined for the line graph of any tree, total graph of any tree, complement of any tree, sum of two non-trivial trees and Mycielskians of some families. For any pair of positive integers a, b ≥ 3 with a ≤ b, there exists a graph whose Steiner radial number is a and Steiner radial number of its line graph is b.
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PDFDOI: http://dx.doi.org/10.5614/ejgta.2025.13.1.3
References
M. Behzad, A criterion for the planarity of the total graph of a graph, Math. Proc. Cambridge Philos. Soc. 63 (1967), 679–681.
M. Behzad, The connectivity of total graphs, Bull. Aust. Math. Soc. 1 (1969), 175–181.
M. Behzad and H. Radjavi, The total group of a graph, Proc. Amer. Math. Soc. 19 (1968), 158–163.
M. Behzad and H. Radjavi, Structure of regular total graphs, J. Lond. Math. Soc. 44 (1969), 433–436.
S.H. Bertz, Branching in graphs and molecules, Discrete Appl. Math. 19 (1998), 65–83.
S.H. Bertz and W.F. Wright, The graph theory approach to synthetic analysis: Definition and application of molecular complexity and synthetic complexity, Graph Theory Notes New York 35 (1998), 32–48.
B. Wang, A. Li, H. Li, and Y. Chen. Graphfl, A federated learning framework for semi-supervised node classification on graphs, arXiv preprint arXiv:2012.04187, 2020.
F. Buckley and F. Harary, Distance in graphs, Addison-Wesley, Reading, 1990.
G. Chartrand, O.R. Oellermann, S. Tian. and H.B. Zou, Steiner distance in graphs, Casopis Pro Pestovani Matematiky 114 (4) (1989), 399–410.
A.A. Dobrynin and L.S Melnikov, Wiener index of generalized stars and their quadratic line graphs, Discuss. Math. Graph Theory 26 (1) (2006), 161–175.
I. Gutman, L. Popovic, B.K. Mishra, M.Kaunar, and N.Guevara, Appliccation of line graphs in physical chemistry. Predicting surface tension of alkanes, Journal of the Serbian Chemical Society 62 (1993), 1025–1029.
I. Gutman and E. Estrada, Topological indices based on the line graph of the molecular graph, Journal of Chemical Information and Computer Sciences 36 (1996) 541–543.
D.A. Holton, D.Lou, and K.L.McAvaney, n-Extendability of line graphs, power graphs and total graphs, Australas. J. Combin. 11 (1995), 215–222.
K.M. Kathiresan, S.Arockiaraj, R. Gurusamy, and K.Amutha, On the Steiner radial number of graphs, In IWOCA 2012, S.Arumugam and W.F. Smyth (Eds.), Springer-Verlag, Lecture Notes in Comput. Sci. 7643 (2012), 65–72.
K.M. Kathiresan and G. Marimuthu, A study on radial graphs, Ars Combin. 96 (2010), 353–360.
J. Mycielski, Sur le coloriage des graphes, Colloquium Mathematicum 3 (1955), 161–162.
O.R. Oellermann and S. Tian, Steiner centers in graphs, J. Graph Theory 14 (5) (1990), 585–597.
O. Russakovsky and A.Y. Ng, A Steiner tree approach to efficient object detection, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, San Francisco, CA, USA, (2010), 1070–1077, doi: 10.1109/CVPR.2010.5540097.
L. Schmidt, C. Hegde, P. Indyk, L. Lu, X. Chi, and D. Hohl, Seismic feature extraction using Steiner tree methods, 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), South Brisbane, QLD, Australia, (2015), 1647–1651, doi: 10.1109/ICASSP.2015.7178250.
Susilawati, E.T. Baskoro, and R. Simanjuntak, Total vertex irregularity strength of trees with maximum degree five, Electron. J. Graph Theory Appl. 6 (2) (2018), 250–257.
Y. Mao, Steiner Distance in Graphs—A Survey, arXiv:1708.05779, (2017), 1–85.
Y. Hafidh and E.T. Baskoro, Partition dimension of trees - palm approach, Electron. J. Graph Theory Appl. 12 (2) (2024), 265–272.
Z. Zheng, Y. Zhou, Y. Sun, Z. Wang, B. Liu, and K. Li, Applications of federated learning in smart cities: recent advances, taxonomy, and open challenges, Connection Science 34 (1) (2022), 128.
Z. Wu, S. Pan, F. Chen, G. Long, C Zhang, and S.Y. Philip, A comprehensive survey on graph neural networks, IEEE transactions on neural networks and learning systems, 32 (1) (2020), 424.
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