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For an odd prime number p, and positive integers k and
, we denote
, a digraph for which
is the set of vertices and there is a directed edge from
*u* to
*v* if
, where
. 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
, then all fixed points in
are isolated. It is proved that the digraph
has
isolated fixed points only if
. It has been characterized that
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
has been discussed.

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

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 [

Let

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 [

Definition 1. (see [

Theorem 0. (see [

1. Let g be a quadratic residue of q, then

2. A point t is fixed Û

3. Fixed points of f are multiples of the order of g.

4. Let

Let’s draw a digraph of the Lambert map. Take

Recall that a vertex u is said to have a loop ( fixed point) on it if

Lemma 1. Let p be any prime. Then,

Proof. Let

,

The proof of the following theorem is simple and can be established similar to Theorem 0 (4).

Theorem 1. Let

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

Proof. Let

For the rest of the proof, we note that

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

Proof. Let

Thus

Lemma 4. Let

Proof. Let

Theorem 2. If

Proof. Let

Theorem 3. Let

i) If

ii) 0 is an isolated fixed point of G if and only if

iii) If

Proof. i) Let

This means that either

ii) Let

Conversely, suppose 0 is isolated. Let there be any integer k such that

iii) Let

This shows that

The following corollaries are the simple consequences of above theorem.

Corollary 1. Let

Corollary 2. If

Theorem 4. The digraph

Proof. By Lemma 1,

In

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

The following result tackle this case and enumerate the solutions as well. Note that the vertex

Theorem 5. Let p be an odd prime and

1. If

2. Let

In particular,

3. If g is a primitive root of

Thus, 0 and

4. If

We are very thankful to the editor and the reviewers for specially sparing their precious time and forwarding useful comments. We sincerely believe that this has made the manuscript more interesting and informative.

M. Khalid Mahmood,Lubna Anwar, (2016) Loops in Digraphs of Lambert Mapping Modulo Prime Powers: Enumerations and Applications. Advances in Pure Mathematics,06,564-570. doi: 10.4236/apm.2016.68045