A Moment-Based Method of Optinalysis

Graph symmetry detection, similarity, and identity measures have been extensively studied in graph automorphism and isomorphism problems. Nevertheless, graph isomorphism and automorphism detection remain an open (unsolved) problem for many decades. In this paper, a new optinalytic coefficient termed as an optical moment coefficient was introduced for optinalysis. Its characteristic efficiency was tested for bijective property, invariance, deterministic polynomiality and non-polynomiality. The test results show that the new optical moment coefficient is very efficient for symmetry detections, similarity and identity measures between two isometric isomorphs and automorphs; and deterministic on polynomial and nonpolynomial graph models.


Introduction
The notion of isometry (as a congruence mapping) is a general phenomenon commonly accepted in Mathematics. It means a mapping that preserves distances. It is a bijective mapping, characterized as one-to-one mapping of a group onto itself or onto another in various transformational ways such as reflections, translation, or rotations (Miillman, R & Parker, G, 1981).
Two graphs or sets are isomorphic if there is a bijection between their vertices or elements that preserves adjacency; such a bijection is called an isomorphism. In other terms, two graphs or sets A and B as isomorphic if they have the same structure, but their elements or vertices may be different. An isomorphism from a graph onto itself is called an automorphism, and the set of all automorphisms of a given graph G denoted Aut(G), forms a group under composition (Hatori et al., 2006;Joyce, 2015).
The graph isomorphism problem is that of detecting the existence (or not) of an isomorphism between two input graphs. It is among a notorious problem in complexity theory, as no polynomial-time algorithm is known, and at the same time graph isomorphism is generally not believed to be NP-complete (Laszlo, 2016). Babai recently presented a quasipolynomial time algorithm for graph isomorphism (Oded et al., 1991). It is well-known that graph isomorphism is algorithmically and computationally polynomialtime equivalent (Ronald, C, and Derek, G 1977;Mathon 1979).
In applications, graph isomorphism has been applied to many structural entities from molecules to objects, and from music to physics, etc. For instance, several methods for detection of isomorphism problems in Kinematics such as Linkage Characteristic Polynomials Method, Hamming Number Technique, Degree Code Method, Link Adjacency Table Method, Distance Concept, Neural Network Approach, Fuzzy Logic Approach, Loop Based Detection Method, Genetic Algorithm Approach, Spanning Tree Method, Adjacency Matrix Method, Joint-Joint Matrix Method, etc have been proposed by many scholars from machine learning and algorithmic community (Arora & Nigam, S, 2013;Shane et al., 2014). In addition to the polynomiality formalization, reliability of results, simplicity, applicability, detection of inversions, etc are some of the definable attributes of efficient methods or algorithms in graph isomorphism detections (Arora & Nigam, S, 2013;Shane et al., 2014). Until date, none of the existing methods is generally accepted to be efficient and a complete solution.
In this paper, an optical moment coefficient of optinalysis is introduced which looks at two finite graphs and sets as isometric isomorphs or automorphs as a mirror-like reflection of each other that expresses their degree of symmetry or identity and similarity. Optical moment coefficient of optinalysis is expressed in clearly defined terms and tested under some theorems and definitions of isometry, isomorphism, automorphism and optinalysis.

Preliminary definitions and theorems
Definition I. Injections, surjections, and bijections of functions between sets (Joyce, 2015).
These are words that describe certain functions ∶ → →Bfrom one set to another.
An injection, also called a one-to-one function is a function that maps distinct elements to distinct elements, that is, if x ≠ y, then f(x) ≠ f(y). Equivalently, if f(x) = f(y) then, x = y.
A surjection also called an onto function is one that includes all of B in its image, that is, if ∈ , then there is an ∈ such that ( ) = .
A bijection, also called a one-to-one correspondence, is a function that is simultaneously injective and bijective. Another way to describe a bijection ∶ → is to say that there is an inverse function ∶ → so that the composition ∶ → is the identity function on A while ∶ → is the identity function on B. The usual notation for the function inverse to is .
If and are inverse to each other, that is, if is the inverse of , = , then is the inverse of , = Thus, ( ) = .
An important property of bijections is that you can convert equations involving to equations involving : ( ) = if and only if = ( ).
Definition II. Isometry (or congruence or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. Let and be metric space with metrics and . A map ∶ → is called an isometry or distance preserving if for any , ∈ one has

Definition III.
Isomorphism is a vertex bijection that preserves the adjacency or mathematical structures (e.g, vertices, edges, and non-edges) between two spaces sets and graphs that can be reversed by inverse mapping. Two mathematical structures A and B are isomorphic if they have the same structure, but their elements may be different (Hatori et al., 2006;Joyce, 2015).
∶ → ≅ Definition IV. Automorphism is an isomorphism from a mathematical object to itself. It is, in some sense; define the symmetry of the object, and a way of mapping the object to itself while preserving all of its mathematical structure (e.g vertices, edges, and non-edges) (Hatori et al., 2006;Joyce, 2015).
Scale can be defined as the system of marks at fixed intervals, which define the relationship between the units being used and their representation on the graph.

Definition VI.
Optinalysis is a function that automorphically or isomorphically compares the symmetry, similarity, and identity between two spaces, graphs, and sets as a mirror-like (optic-like) reflection of each other about a symmetrical line or mid-point (Abdullahi, 2020).

Definition VII.
Optinalysis is a function that comprised of an assigned optical (mirror) scale (R) that bijectively re-maps (↠) isometric and isomorphic or automorphic spaces, sets, and graphs under reflection about a symmetrical line or mid-point ( ). Figure 1 illustrates how two isometric isomorphs or automorphs are mapped and also re-mapped by an optical scale (Abdullahi, 2020).

Definition VIII.
In comparative optinalysis, a reflection (pairing) is said to be head-to-head if the first labeled elements of two graphs, sets are the terminal points of the reflecting isometric isomorphs (Abdullahi, 2020). Theorem I. Isometric and isomorphic or automorphic spaces, sets and graphs are symmetrical or identical and similar to a certain magnitude by a coefficient (Abdullahi, 2020).

Optinalytic constructions in general sense in the new optical scale
Following this definition of optinalysis in box:

Definition
Optinalysis is a function that comprised of an assigned optical (mirror) scale (R) that bijectively re-maps (↠) isometric and isomorphic or automorphic spaces, sets, and graphs under reflection about a symmetrical line or mid-point ( ). Figure 1 illustrates how two isometric isomorphs or automorphs are mapped and also re-mapped by an optical scale (Abdullahi, 2020).
We can establish the optinalytic constructions as in Figure 1, which illustrates how two isometric isomorphs or automorphs are bijectively mapped and also bijectively re-mapped by the new optical scale. is a domain; is a co-domain; is a mid-point or symmetrical line, and is an optical scale. The symbol ⇻ indicates a bijective mapping between the isometric isomorphs around a centre (midpoint) and ↠ indicates a bijective re-mapping by the optical scale.

The optinalysis construction in isomorphic optinalysis
It is expressed as: ≠ 0; and & are isometric isomorphs of a named space, sets, and graphs.

Definitions in Moment-Based Method of Optinalysis
Optinalysis is a function that bijectively maps optical nodes ( ) to their corresponding optical nodes ( ′) about a symmetrical line or mid-point ( ). An optical node is defined by its moment. A moment, is the product of any member of a set and its assigned optical scale.

Definition by Automorphic Optinalysis
Suppose we have an optinalytic construction of isometric and automorphic spaces, sets and graphs with an assigned optical scale (R = 3, 2, 1, 0, 1, 2, 3) as follows: Such that and ′ are isometric isomorphs on head-to-head reflection about a centre ( ). :

Definition by Isometric Optinalysis
Suppose we have an optinalytic construction of isometric and isomorphic spaces, sets and graphs with an assigned optical scale (R = 3, 2, 1, 0, 1, 2, 3) as follows: Such that and are isometric isomorphs on head-to-head reflection about a centre ( ). :

Optical moment coefficient of optinalysis
An optical moment coefficient is expressed as the ratio of the sum of lowest optical moment to the sum of highest optical moment about a symmetrical line or midpoint. A valid result lies within a range ≥ 0 ≤ 1.

Optical moment coefficient by shape optinalysis
Suppose we have an optinalytic construction of isometric and automorphic spaces, sets and graphs with an assigned optical scale (R = 3, 2, 1, 0, 1, 2, 3) as follows: Such that and ′ are isometric automorphs on head-to-head reflection about a centre ( ).
Then, the Optical moment coefficient of symmetry or identity and similarity between the two isometric isomorphs is expressed as (Equations 1) if ∈ , then there is an ∈ such that ( ) = if ∈ , then there is an ∈ such that ( ) = if ∈ , then there is an ∈ such that ( ) = Therefore, optinalysis is based on the paradigm of the bijective mapping of defined finite sets of isometric isomorphs or automorphs.

The test for optinalytic invariance (I)
Optinalysis theorem: A perfect symmetry or identity and similarity state between isometric and isomorphic groups remains invariant (stable) with = 1 under other transformations such as pericentral rotation (alternate reflection), central rotation (inversion), product translation, additive translation, optical scaling (Abdullahi, 2020).
Suppose we have an optinalytic construction of isometric and isomorphic spaces, sets and graphs with an assigned optical scale (R = 3, 2, 1, 0, 1, 2, 3) as follows: Therefore, and are perfectly similar and identical.

The test for optinalytic invariance under additive translation
Let the optinalytic construction of prove 3 be considered, and let ′ ′ be a translation factor. The optinalytic construction becomes: Therefore, and are invariant under translation.

The test for optinalytic invariance under product translation
Let the optinalytic construction of prove 3 be considered, and let ′ ′ be a translation factor. The optinalytic construction becomes: Such that and are isometric isomorphs on head-to-head reflection about a centre ( ). Then, Therefore, and are invariant under translation.

The test for optinalytic invariance under central rotation (Inversion)
Let the optinalytic construction of prove 3 be considered, the centrally rotated structures becomes: Such that and are isometric isomorphs on head-to-head reflection about a centre ( ). Then, Therefore, and are invariant under central rotation.

The test for optinalytic invariance under pericentral rotation (Alternate reflection)
Let the optinalytic construction of prove 3 be considered, the pericentrally rotated structures (inverses) of isometric and isomorphs becomes: Such that and are isometric isomorphs on tail-to-tail (its alternate) reflection about a centre ( ). Then, Therefore, and are invariant under pericentral rotation.

The test for optinalytic invariance under optical scaling
Let the optinalytic construction of prove 3 be considered, and let + be the change in scaling patterns. The optinalytic construction becomes: Therefore, and are invariant under optical scaling.

The test for optinalytic invariance under operations (II)
Optinalysis theorem: Asymmetrical or dissimilar state between isometric and isomorphic or automorphic remains invariant (the same) under product translation, central rotation (inversion), and optical scaling (Abdullahi, 2020).

The test for optinalytic invariance under product translation
Let the optinalytic construction of prove 4 be considered, and let ′ ′ be a translation factor. The optinalytic construction becomes: Such that and are isometric isomorphs on head-to-head reflection about a centre ( ). Then, Therefore, and are invariant under product translation.

The test for optinalytic invariance under central rotation (Inversion)
Let the optinalytic construction of prove 4 be considered, the centrally rotated structures of isometric isomorphs become: Therefore, and are invariant under product translation.

The test for optinalytic invariance under optical scaling
Let the optinalytic construction of prove 4 be considered, and let (− − ) be the change in scaling patterns. The optinalytic construction becomes: Therefore, and are invariant under optical scaling.

The test for optinalytic inverse equivalence
Optinalysis theorem: The optical moment coefficients of optinalysis for inverse isometric isomorphs or automorphs remain the same as its former (Abdullahi, 2020).
Suppose we have an optinalytic construction as follows: Now, the inverse equivalence by optical moment coefficient is Therefore, the inverse equivalences by percentage (probability) are equals to each other.

The test for deterministic polynomiality and non-polynomiality
Optinalytic theorem: Similarity, identity, and symmetry of isometric isomorphs and automorphs from named spaces, sets, and graphs are deterministic by optical moment coefficient on polynomial and nonpolynomial graph models, and also by the translated percentages or probabilities on polynomial and nonpolynomial graph models (Abdullahi, 2020).
Then, the optical moment coefficient between the graph and its iso-polymorphs ( → ) were obtained as Q.
Let G be a set of positive integers that ranks the iso-polymorphic generations (from the first to the last) of iso-polymorphs established.
By plotting regression graphs of G against Q, we observe a moving and changing regression patterns (from the best fits of linear/polynomial/exponential, continues as polynomial, shifts to logarithmic, and finally stagnates at power) as a skewization value approaches a certain maximum (See Fig. 2).

Results
The results present in Figures 2 reflect and justify the deterministic polynomiality and non-polynomiality of optinalytic theorem.

Discussion
In this paper, a moment-based method of optinalysis for symmetry detections, similarity, and identity measures between two established isometric and isomorphic or automorphic graphs and sets is introduced. When we look at the optical scale, its uniform intervals preserve an equidistant relationship between the corresponding vertices or nodes of a named space, set, or graph. Furthermore, the optinalytic relationship between any pair of corresponding vertices is a clear bijection, which is also invertible, and these conform and translate the definitions and theorems of isometry, isomorphism, and automorphism.
When we look at the optical moment coefficient function, we tested that the results remain the same (invariant) under mathematical operations or transformations. This is a preservation of the mathematical structures of the isomorphs or automorphs. However, these properties of optical moment coefficient of optinalysis are sufficient evidences to prove its efficiency in graph symmetry detection, graphs similarity, and identity measures.
Concerning methodological efficiency, the polynomial and non-polynomial graph models generated by optical moment coefficient of optinalysis are very much the same as one of those generated by Kabirian coefficient of optinalysis (Abdullahi, 2020), and also very much analogous to the algorithmic formalization of polynomial-time of the effectiveness of computational methods of graph isomorphism problem. Therefore, optical moment coefficient of optinalysis is another complete solution to the graph isomorphism and automorphism problems in relation to symmetry detections, similarity and identity measures. This solution is analogous to Kabirian coefficient of optinalysis (Abdullahi, 2020), and unlike the recent quasipolynomial algorithm introduced by Babai (Laszlo, 2016).

Conclusion
Optical moment coefficient of optinalysis is an efficient moment-based method for symmetry detections, similarity, and identity measures between two established isometric and isomorphic or automorphic graphs and sets. The optical moment coefficient of optinalysis is based on isometry, isomorphism, and automorphism, expressed in clearly defined terms is tested to conform to the definitions and theorems of isometry, isomorphism, automorphism and optinalysis. Analogous to the polynomiality