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STRUCTURES AND METRIC DIMENSIONS OF DIVISOR EULER FUNCTION GRAPH $G(D(\phi(t))$
Keywords:
Keywords and phrases: graph of divisor function $D(t)$, Euler function graph $G(\phi(t))$, divisor Euler function graph $G(D(\phi(t)))$, divisor Euler function sub-graph $H(D(\phi(t)))$, resolvent $\mathfrak{R}$, metric dimension.DOI:
https://doi.org/10.17654/0974165825022Abstract
The symbol phi(t) counts positive integers less than $t$ that are coprime to $t$, whereas $\tan (t)$ represents the number of divisors of $t$. We construct a new number theoretic graph called Divisor Euler Function Graph (DEFG), labeled as $G(D(\phi(t)))$, by incorporating $p h(t)$ and $\operatorname{tau}(t)$. The motivation for this work was to discover the structures of these newly defined graphs (DEFG), followed by an attempt to discover the metric dimensions of such graphs. In $G(D(\phi(t)))$, the vertex set $\mathbb{V}$ assumes divisors of $t$, where the edge set $\mathbb{E}$ is based on $\phi(t)$. The metric representation of any arbitrary vertex $v$ with respect to an ordered subset $\Re=\left\{d_1, d_2, \ldots, d_k\right\}$ of $\mathbb{V}$, is the $k$-vector, written as, $r(v \mid R e)=\left(d\left(v, d_1\right), d\left(v, d_2\right), \ldots, d\left(v, d_k\right)\right)$ such that $d\left(v, d_i\right)$ is the shortest distance between the vertices $v$ and $d_i$. In this piece of work, we discuss structures of DEFG such as loops, cycles and their lengths, connectedness, maximal and minimal degree of vertices, components of complete graphs as bipartite graphs, planarity, Hamiltonicity, and Eulerianity. We further find chromatic number, chromatic index and clique numbers of these graphs. Also, we compute distance codes, resolving sets, local resolving sets, metric dimension and local metric dimension of certain families in DEFG.
Received: October 8, 2024
Revised: November 22, 2024
Accepted: December 2, 2024
References
M. Khalid Mahmood and Shahbaz Ali, On super totient numbers, with applications and algorithms to graph labeling, Ars Combinatoria 143 (2019), 29-37.
S. Shanmugavelan, The Euler function graph Int. J. Pure App. Math. 116 (2017), 45-48.
J. Baskar Babujee, Euler’s phi function and graph labeling, Int. J. Contemp. Math. Sci. 5 (2010), 977-984.
M. Manjuri and B. Maheswari, Matching dominating sets of Euler totient Cayley graphs, Int. J. Comp. Eng. Res. 2(7) (2012), 104-107.
J.-B. Liu, M. F. Nadeem, H. M. A. Siddiqui and W. Nazir, Computing metric dimension of certain families of Toeplitz graphs, IEEE Access 7 (2019), 126734-126741.
Futaba Okamoto, B. Phinezy and P. Zhang, The local metric dimension of a graph, Mathematica Bohemica 135(3) (2010), 239-255.
Min Feng, Benjian Lv and Kaishun Wang, On the fractional metric dimension of graphs, Discrete Applied Mathematics 170 (2014), 55-63.
Jia-Bao Liu, Muhammad Kamran Aslam and Muhammad Javaid, Local fractional metric dimensions of rotationally symmetric and planar networks, IEEE Access 8 (2020), 82404-82420.
K. Kannan, D. Narasimhan and S. Shanmugavelan, The graph of divisor function Int. J. Pure App. Math. 102(3) (2015), 483-494.
M. H. Mateen and M. Khalid Mahmood, Power digraphs associated with the congruence Punjab Univ. J. Math. 51 (2019), 93-102.
M. Khalid Mahmood and Shahbaz Ali, A novel labeling algorithm on several classes of graphs, Punjab Univ. J. Math. 49 (2017), 23-35.
F. Harary and R. A. Melter, The metric dimension of a graph, Ars Combinatoria 2 (1976), 191-195.
M. H. Mateen and M. Khalid Mahmood, A new approach for the enumeration of components of digraphs over quadratic maps, J. Prime Res. Math. 16 (2020), 56-66.
M. H. Mateen, M. Khalid Mahmood, S. Ali and M. D. Alam, On symmetry of complete graphs over quadratic and cubic residues, Journal of Chemistry (2021), 10.
M. H. Mateen, M. Khalid Mahmood, D. Alghazzawi and J. B. Liu, Structures of power digraphs over the congruence equation and enumerations, AIMS Math. 6(5) (2021), 4581-4596.
M. Farooq, A. Abd ur Rehman, M. Khalid Mahmood and Daud Ahmad, Upper bound sequences of rotationally symmetric triangular prism constructed as Halin graph using local fractional metric dimension, VFAST Trans. Math. 9(1) (2021), 13-24.
T. Sabahat, S. Asif and A. Abd ur Rehman, Structures of digraphs arizing from Lambert type maps, VFAST Trans. Math. 9(1) (2021), 28-36.
T. Sabahat, S. Asif and A. Abd ur Rehman, On fixed points of digraphs over Lambert type map, VFAST Trans. Math. 9(1) (2021), 59-65.
D. Narasimhan, R. Vignesh and A. Elamparithi, Directed divisor function graph Int. J. Eng. Adv. Tech. (IJEAT) 8 (2S) (2018).
S. Shanmugavelan, K. T. Rajeswari and C. Natarajan, A note on indices of primepower and semiprime divisor function graph, TWMS J. App. Eng. Math. 11(SI) (2021), 51-62.
John Rafael M. Antalan, Jerwin G. De Leon and Regine P. Dominguez, On k dprime divisor function graph, 2021. arXiv:2111.02183v2.
Jelena Sedlar and Riste Skrekovski, Vertex and edge metric dimensions of unicyclic graphs, Discrete Appl. Math. 314 (2022), 81-92.
G. Chartrand, L. Eroh, M. A. Johnson and O. R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math. 105 (2000), 99-113.
G. S. Singh and G. Santhosh, Divisor Graphs-I, Preprint, 2000.
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