Loops in Digraphs of Lambert Mapping Modulo Prime Powers: Enumerations and Applications

Abstract

For an odd prime number p, and positive integers k and

The Iteration Digraphs of Lambert Map Over the Local Ring $mathbb{Z}/p^kmathbb{Z}$ : Structures and Enumerations

“…By Theorem 2.7, G(g, p k ) contains p k +1 2 fixed points. Also by Lemma 1 [9], ord p k g = 2. Hence by Proposition 3.1, there exists no cycle of length > 1.…”

“…Suppose p ∤ g but g 2 ≡ 1 (mod p). Since g = 1 be any integer, so by Lemma 1 [9], g = p k − 1. Then by Theorem 2.3 (4), if t is odd and g = p k 1 then f (t) = p k −t and if t is even then f (t) = t. Suppose that there exists a cycle of length 2.…”

Structures of Digraphs Arizing from Lambert Type Maps

“…In 1993 it was revealed that the Lambert function gives an exact solution to the quantum-mechanical double-well Dirac delta function model for equal charges [3]. In 2016 Khalid and Lubna proposed loops in digraphs of Lambert Mapping modulo of prime powers and its applications [4].…”

“…d. d ≡ t4 (mod r e ). Now we must demonstrate that ḋ ≡ (-1) m d (mod r e ),ḋ ≡ m 2 (-h) m (mod r e )(31)≡ m 2 (-1) m h m (mod r e )(32)≡ (-1) m m 2 h m (mod r e ) (33)≡ (-1) m d (mod r e ).…”

Upper Bound Sequences of Rotationally Symmetric Triangular Prism Constructed as Halin Graph Using Local Fractional Metric Dimension

“…Further, Currie and Oellarmann introduced the concept of fractional metric dimension for getting more accurate solution of integer programming problem [12]. In [13][14][15][16][17][18][19][20][21][22][23][24][25][26], authors have already proposed several classes of integers based on partitions of integers together with graph structures and their applications in graph labeling. In this paper, we propose and prove generalized formulas of all sequences for local fractional metric dimension over triangular prism.…”