Symmetry, as a key concept in geometry and the foundational principle of mathematical concepts, is a fundamental part of geometry and nature, creating patterns that help us conceptually organize our world. It is often linked with aesthetic beauty and can be observed in both natural phenomena and human-made objects, such as artworks and manufactured products, across the globe. The automorphism group of a graph is a mathematical structure that captures all the symmetries of the graph. symmetry refers to the property of a graph, indicating that it can be transformed without changing its structure, while the automorphism group is the collection of all permutations of the vertices that preserve the graph's adjacency relationships. In graph theory, the concepts of symmetry group and automorphism group play fundamental roles in understanding the symmetries and structural properties of graphs. This paper aims to explore the relationship between symmetry groups and automorphism groups, shedding light on their interconnected nature and their significance in graph theory. We delve into the definitions and properties of both symmetry groups and automorphism groups, highlighting their similarities and distinctions. Furthermore, we discuss their applications in various fields, including computer science, chemistry, and network analysis. Through this exploration, we aim to provide a comprehensive understanding of the relationship between symmetry groups and automorphism groups, deepening our comprehension of the symmetries and structures inherent in graphs.

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 non-polynomial graph models.

Measures of graph symmetry, similarity, and identity have been extensively studied in graph automorphism and isomorphism detection problems. Nevertheless, graph isomorphism detection remains an open (unsolved) problem for many decades. In this paper, a new and efficient methodological paradigm, called optinalysis, is proposed for symmetry detections, similarity, and identity measures between isometric isomorphs or automorphs. Optinalysis is explained and expressed in clearly stated definitions and prove theorems, which conform to the definitions and theorems of isometry, isomorphism, and automorphism. Analogous to the polynomiality formalization for an efficient algorithm for graph isomorphism detection, optinalysis is however deterministic on polynomial and non-polynomial graph models.

In this paper we present two ways to solve the Jacobian conjecture. The first way is an equivalent statement to the Jacobian conjecture using idempotent ideals. The second way is to use an equivalent thesis to the thesis of the Jacobian conjecture. In both cases we will show that the Jacobian conjecture is true.

Suppose G is a graph with vertex-orbits O1,1 O2, . . . ,Ot, and jOi j denotes the cardinallity of Oi. Then OG (x) = åt 2 i=1 xjOi j is called as orbit polynomial. It is well-known that this polynomial 3 has a unique positive zero d in the interval [0, 1]. The aim of this paper is to study the specific 4 properties of this polynomial and then we determine the location of this root for several classes of 5 complex networks to compare with other graphical measures. Besides, we compare the unique 6 positive zero measure with several well-known centrality graph measures.

This paper describes a polynomial time algorithm for solving graph isomorphism and automorphism. We introduce a new tree data structure called Walk Length Tree. We show that such tree can be both constructed and compared with another in polynomial time. We prove that graph isomorphism and automorphism can be solved in polynomial time using Walk Length Trees.

The Svenonius theorem establishes the correspondence between definability of relations in a countable structure and automorphism groups of these relations in the extension of the structure. This correspondence may help in finding the description of the definability lattice constituted by all definability spaces (reducts) of the original structure. However, the major difficulty here is the necessity to consider the extensions which generally are obscure and hardly amenable to classification. Because of that results on definability lattices were obtained only for $\omega$-categorical structures (i.\,e. those in which all elementary extensions are isomorphic to the structure itself). All known definability lattices for such structures proved to be finite. In this work we introduce the concept of upwards complete structure as such in which all its extensions are isomorphic. Further we define upwards completions structures. For such structures Galois correspondence between definability lattice and the lattice of closed supergroups of automorphism group of the structure is an anti-automorphism. These lattices could be infinite in general. We describe the natural class of structures which have upwards completion, that is discretely homogeneous graphs, present the explicit constrcution of their completion and automorphism groups of completions. We establish the general \emph{localness} property of discretely homogeneous graphs and present the examples of completable structures and their completions.

Let N = N₀+ N₁+ N₂ be a Z₃-graded Novikov algebra. The main goal of the paper is to prove that over a field of characteristic not equal to 3 the algebra N is solvable if N₀ is solvable. We also show that a $Z_2$-graded Novikov algebra N=N₀+ N₂ over a field of characteristic not equal to 2 is solvable if N₀ is solvable. This implies that for every n of the form n=2^k3^l, any Z_n-graded Novikov algebra N over a field of characteristic not equal to 2,3 is solvable if N₀ is solvable.

Using the orbit decomposition, a new enumerative polynomial P(x) is introduced for abstract (simplicial) complexes of a given type, e.g., trees with a fixed number of vertices or triangulations of the torus with a fixed graph. The polynomial has the following three useful properties. (I) The value P(1) is equal to the total number of unlabeled complexes (of a given type). (II) The value of the derivative P'(1) is equal to the total number of nontrivial automorphisms when counted across all unlabeled complexes. (III) The integral of P(x) from 0 to 1 is equal to the total number of vertex-labeled complexes, divided by the order of the acting group. The enumerative polynomial P(x) is demonstrated for trees and then is applied to the triangulations of the torus with the vertex-labeled complete four-partite graph G = K_{2,2,2,2}, in which specific case P(x) = x^{31}. The graph G embeds in the torus as a triangulation, T(G). The automorphism group of G naturally acts on the set of triangulations of the torus with the vertex-labeled graph G. For the first time, by a combination of algebraic and symmetry techniques, all vertex-labeled triangulations of the torus (twelve in number) with the graph G are classified intelligently without using computing technology, in a uniform and systematic way. It is helpful to notice that the graph G can be converted to the Cayley graph of the quaternion group Q_8 with the three imaginary quaternions i, j, k as generators.