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

For an odd prime number p, and positive integers k and ( ) k g p   / ∗ ∈ , we denote ( ) k G g p , , a digraph for which { } k p 0,1, 2, , 1 −  is the set of vertices and there is a directed edge from u to v if ( ) ( ) k f u v mod p ≡ , where ( ) x f x xg = . In this work, we study isolated and non-isolated fixed points (or loops) in digraphs arising from Discrete Lambert Mapping. It is shown that if ( ) k i g mod p i k 1 ,1 1 − ≡ ≤ ≤ − , then all fixed points in ( ) k G g p , are isolated. It is proved that the digraph ( ) k G g p , has k p 1 − isolated fixed points only if k g tp t p 1 1, 1 1 − = + ≤ ≤ − . It has been characterized that ( ) k G g p , has no cycles except fixed points if and only if either g is of order 2 or g is divisible by p. As an application of these loops, the solvability of the exponential congruence ( ) x k xg x mod p ≡ has been discussed.


Introduction
The Lambert W functions are used to find solutions of such equations in which the unknown also appears in exponential (or logarithmic) terms.It is defined as c W c = , where c is a complex number.Equivalently, it can be defined as ( ) e w f w w = .Lambert solved a Diophantine equation k x x t = + in 1758 (see [1]).Later, the solution is expressed in term of series.In 1980, the Lambert function was stored in MCAS (Maple Computer Algebra System) as a function for the solution of algebraic equations involving exponential (or logarithmic) functions (see [2]).In this work, we discussed solutions of such functions by means of their digraphs using residue theory from number theory.
Let  be the ring of residue classes modulo k p .Define : f    by ( )  .We investigate this mapping using directed graphs whose vertices are residues modulo k p with edges from u to v if and only if ( ) ( ) . This digraph is denoted by ( ) We investigate self loops (fixed points) of these digraphs and also lift up the investigations of such digraphs by Jingjing Chen and Mark Lotts in [3] from modulo a prime p to modulo k p .Results regarding fixed points, isolated points followed by astute proofs have been presented.It is important to note that all solutions of congruences of Lambert functions are difficult to find since such mappings are hard to invert and need enormous inversions in any computer algorithm.To understand the terminology and symbols, we follow [3]- [6].
Definition 1. (see [7]).Let p be prime and a be any integer not divisible by p.A least positive integer r such that ( ) Theorem 0. (see [3]).Let q be any prime and Let g be a quadratic residue of q, then ( ) Fixed points of f are multiples of the order of g. 4. Let 1 g q = − .If t is odd, then ( ) f t q t = − , and if t is even, then ( ) = is a fixed point.Let's draw a digraph of the Lambert map.Take 13 g = and chose a composite modulus m as 15 3 5 m = = × .We see that the digraph (see Figure 1) has six loops (fixed points) of which three are non-isolated.The digraph has two non-isomorphic components.

Fixed Points of the Map
. This means that either ( ) . But the first implies that, Ord 1 k p g = .Hence ( ) p then a a  under f.In the following theorem, we find the values of g for which the fixed points of the digraph are necessarily isolated.Before proving the assertion, we give the following important lemmas.

Lemma 2. If
( ) > are the fixed points of the graph ( ) , k G g p .In particular, the vertices, for some integer s.But then For the rest of the proof, we note that 1 1 2 2 The case when k is odd can be dealt in a similar technique. The following Lemma is of crucial importance.However, its proof is simple and can be viewed as a direct consequence of the Definition 1.
Lemma 3. Let g be a residue of k p .Then ord k i p g p = if and only if .
Then l is the least positive integer such that ( ) ( ) for some integer s such that ( ) ( ) if and only if ( ) Then α is the least positive integer such that ( ) ≤ ≤ − , for any integer λ , are the fixed points.We need only to show that these are the possible fixed points and are isolated.
, where 1 1 t p ≤ ≤ − and 1 1 i k ≤ ≤ − .Let x be any fixed point in ( )  2 depicts Theorems 2 and 3.In Figure 2, we note that 25 ord 11 5 = .By Theorem 2, the vertices 5, 10, 15, 20 are the fixed points and are isolated.Also 11 g = is not a multiple of 5, so by Theorem 3, 0 is also an isolated fixed point.Thus all fixed points are isolated.
Theorem 3. Let ( ) ( ) ) .This shows that 0 is not isolated, a contradiction.Therefore This shows that tp is a primitive root of k p .But ( ) , k tp p p ≥ .Thus the word primitive root arrive at a contradiction.
 The following corollaries are the simple consequences of above theorem.≡ , being even, a contradiction.Thus there does not exist any cycle of length > 1.Now if g is a multiple of p then it is trivial that all vertices constitute one component.Also by Theorem 3(i), if g is a multiple of p then 0 is the only fixed point.Thus the digraph must be a tree with root at 0. Consequently

Applications
In recent years, studying graphs through different structural environments like groups, rings, congruences has become much captivating and dominant field of discrete mathematics.These assignments are easy to handle most of the mathematics which is integral based.A variety of graphs have been introduced and characterized regarding their structures through this dynamism.By means of congruences one can inspect numerous enthralling topographies of graphs and digraphs.Thus it becomes interesting to demonstrate that every congruence can generate a graph and hence under certain conditions on these graphs, the nature and solutions of congruences can be discussed.In this section, we discuss the solvability of the congruence and enumerate their solutions using the results given in previous section.The non-trivial ( other than 0 x = ) solution of the congruence modulo a single prime p is easy to discuss since every x p < is prime to p.So the congruence .Now if we lift up the modulo from p to its higher powers , 1 k p k > , then the vertices which are not prime to p must not follow the fashion as for 1 k = .
The following result tackle this case and enumerate the solutions as well.Note that the vertex 0 x = is the trivial solution in either case.The proof of the following theorem is simple and can be established using results given in Section 2.
work, we study isolated and non-isolated fixed points (or loops) in digraphs arising from Discrete Lambert Mapping.It is shown that if except fixed points if and only if either g is of order 2 or g is divisible by p.As an application of these loops, the solvability of the exponential congruence ( )

Theorem 5 .
Let p be an odd prime and 1 k > .Then the following hold: 0 b ≠ be any integer.If ord k p g b = then the congruence of the following theorem is simple and can be established similar to Theorem 0 (4).
then all fixed points of