Quotient Energy of Zero Divisor Graphs And Identity Graphs

Applications on color (distance) signless laplacian energy of annihilator monic prime graph of commutative rings

The work is devoted to the consideration of applications of mathematics, in particular, graph theory and group theory in chemistry and physics. Chemistry was one of the first to adopt the ideas of graph theory. Mathematical chemistry is a section of theoretical chemistry that studies the application of mathematics to chemical problems. The possibility of formalizing molecules in the language of structural chemistry using graphs allows one to determine the number of different isomers using graph theory methods. The main tool in mathematical chemistry is mathematical modeling using computer technology. The methods of graph theory are actively used in three areas of chemistry: structural chemistry, chemical kinetics, and chemical physics of polymers. Group theory has found application in many areas of physics. Group theory serves to explain the most important characteristics of atomic spectra. Group theory provides an opportunity for a uniform approach to a large number of problems in solid state physics. Matrix groups and unitary group representations are actively used in these studies. Irreducible group representations are used to classify vibrational and electronic states of a crystal. Methods of group theory are applied to models of problems of classifying electronic states of a crystal. In crystallography, much attention is paid to applications of group theory methods to the study of vibrations of atoms that make up a molecule relative to their equilibrium positions. The results can be useful in encoding information in space communications sessions.

Chemical graph theory is a branch of mathematical chemistry which deals with the non-trivial applications of graph theory to solve molecular problems. Graphs containing finite commutative rings also have wide applications in robotics, information and communication theory, elliptic curve cryptography, physics, and statistics. In this paper we discuss eccentric topological indices of zero divisor graphs of commutative rings Z p 1 p 2 × Z q , where p 1 , p 2 , and q are primes. To enhance the importance of these indices a construction algorithm is also devised for zero divisor graphs of commutative rings Z p 1 p 2 × Z q .

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An important part of the computer science is focused on the links that can be established between group theory and graph theory. Cayley graphs can establish such a link but meet some limitations. This paper introduces a new type of graph associated to a group: the G-graphs. We present an implementation of the algorithm constructing these new graphs. We establish a library of the most common G-graphs, using GAP and the SmallGroups library. We give some experimental results with GAP and we show that many classical graphs are G-graphs.

Let R be a commutative ring with Z*(R) as the set of non-zero zero divisors. The zero divisor graph of R, denoted by Γ(R), is the graph whose vertex set is Z*(R), where two distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we investigate the metric dimension dim(Γ(R)) and upper dimension dim+(Γ(R)) of zero divisor graphs of commutative rings. For zero divisor graphs Γ(R) associated to finite commutative rings R with unity 1 ≠ 0, we conjecture that dim+(Γ(R)) = dim(Γ(R)), with one exception that R ≅ Π 𝕑 2 n {\rm{R}} \cong \Pi {\rm\mathbb{Z}}_2^{\rm{n}} , n ≥ 4. We prove that this conjecture is true for several classes of rings. We also provide combinatorial formulae for computing the metric and upper dimension of zero divisor graphs of certain classes of commutative rings besides giving bounds for the upper dimension of zero divisor graphs of rings.

Group theory, an area in mathematics, has undergone extensive research, but not exhaustively. In group theory, especially symmetric groups have been studied in terms of properties and actions. In this research, two groups of the Symmetric Group (Alternating and Dihedral Groups) are studied in terms of their properties and also their direct product between them. Also, ordered sets are studied in this research where their Cartesian product is looked at. Finally, this research combines the direct product of two symmetric groups (alternating group and dihedral group) and the Cartesian product of two ordered sets (<I>X</I>×<I>Y</I>) by group action. This research therefore focuses on determining the combinatorial properties (transitivity and primitivity), invariants (ranks and subdegrees), and structures (suborbital graphs) of this group action. To accomplish this, orbit-stabilizer theorem is used to compute transitivity, block action concept is applied to determine primitivity, and Cauchy-Frobenius lemma is used to compute ranks and subdegrees. For <i>n</i> ≥ 3, it has been confirmed that the group action is transitive and also imprimitive. The rank of this group action is 6 and the subdegrees are obtained using the formula of Theorem 3.3. This research adds new concepts in group theory which will be useful in other areas like graph theory and also application in real life situations.

Let E(G) be the energy of a graph G, which is defined as the sum of the absolute values of all eigenvalues of G. If G contains no cycle , then it is called a triangle-free graph. In this paper, we investigate the upper bounds of the energy for triangle-free graphs in terms of their matching number and characterize all the extremal graphs attaining the upper bounds. In addition, we establish a relation between graph energy and rank.

The energy of a graph is the sum of the absolute values of its eigenvalues. We propose a new problem on graph energy change due to any single edge deletion. Then we survey the literature for existing partial solution of the problem, and mention a conjecture based on numerical evidence. Moreover, we prove in three different ways that the energy of a cycle graph decreases when an arbitrary edge is deleted except for the order of 4.

In this article, we present the generalized ρ dependent polynomials for the calculations of eccentricity, distance, total distance, and degree-based topological indices of the identity graph of Zρ. This is a thorough work in which we present many topological indices and coindices as ρ dependent polynomials. The polynomials presented here can play a key role in the further development of the theory of topological indies for commutative rings. This paper presents a brand new approach to generalizing the topological indices because instead of the traditional way. A set-theoretic method is introduced here that can be very helpful and game-changing in the field of algebraic graph theory first of all sets of vertices for the identity graph of the commutative ring Zρ are partitioned into various sets, which makes it easier to generalize the degrees, distances, and eccentricities of this graph. This paper presents various results that make it easier to generalize the topologicalindices of the identity graph of Zρ.

Graphs and Algorithms In graph theory, the Helly property has been applied to families of sets, such as cliques, disks, bicliques, and neighbourhoods, leading to the classes of clique-Helly, disk-Helly, biclique-Helly, neighbourhood-Helly graphs, respectively. A natural question is to determine for which graphs the corresponding Helly property holds, for every induced subgraph. This leads to the corresponding classes of hereditary clique-Helly, hereditary disk-Helly, hereditary biclique-Helly and hereditary neighbourhood-Helly graphs. In this paper, we describe characterizations in terms of families of forbidden subgraphs, for the classes of hereditary biclique-Helly and hereditary neighbourhood-Helly graphs. We consider both open and closed neighbourhoods. The forbidden subgraphs are all of fixed size, implying polynomial time recognition for these classes.

Adjacency matrix A(G)=[aij] yields the graph energy, which is equal to the addition of absolute values of the eigenvalues G. This research investigates the energy graph class in terms of another graph class after removing a vertex. After deleting a vertex, a relationship between the energy of a complete graph E[kn ] and energy of a splitting graph E(S’ [kn ]) is discovered.

Let R R be a commutative ring with 1 ≠ 0 1\ne 0 , Z ( R ) Z(R) be the set of all zero-divisors of R R , and n ≥ 1 n \geq 1 . This paper introduces the n n -total graph of a commutative ring R R . The n n -total graph of a commutative ring R R , denoted by n − T ( R ) n-T(R) , is an undirected simple graph with vertex set R R , such that two vertices x , y x, y in R R are connected by an edge if x n + y n x^{n} + y^{n} in Z ( R ) Z(R) . Note that if n = 1 n =1 , then the 1 1 -total graph of R R is the total graph of R R in the sense of Anderson-Badawi’s paper on the total graph of a commutative ring. In this paper, we study some graph properties and theoretical ring structure.

We formulate the relationship of quasi-Laplacian energy of some novel classes of graphs with their corresponding original graphs. The novel graphs in our discussion are the -graph, -graph, -graph, total graph, and their join and corona operations graphs. The whole formulation is based on the relationship between quasi-Laplacian energy and the vertex degrees of the novel graph. It is also noted that quasi-Laplacian energy is closely related with the first Zagreb index, number of vertices and edges of the graph. The exact formulas of quasi-Laplacian energy of novel graphs are obtained in terms of the corresponding quasi-Laplacian energies, the first Zagreb indices, and the number of vertices and edges of the original graphs.

On zero-divisor graphs of small finite commutative rings