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A Cayley graph is a graph of a group that is build based on a specific subset of the group. The vertices of the graph are the elements of the group, and two different vertices, x and y are adjacent if the product of x and inverse of y is in the subset. One type of Cayley graph recently introduced is the prime power Cayley graph in which the subset of the group considered in constructing this graph contains the elements dividing the order of the group. In this paper, we construct all possible prime power Cayley graphs based on the subsets for the cyclic group of order pq where p and q are primes. The results show that there are seven possible prime power Cayley graphs for this group. [ABSTRACT FROM AUTHOR]

Research of graphs associated to finite groups has been an attractive topic and many new graphs have been defined using some properties of finite groups. One of the graphs is the subgroup graph when the subgroup is not normal. For a group 퐺 and a subgroup 퐻 which is not normal, the nonnormal subgroup graph is defined as a directed graph with all elements of 퐺 are vertex set and two different elements 푥 and 푦 are linked from 푥 to 푦 if their product is in 퐻. There are two well-known matrices representing any graph which are the incidence matrix and the adjacency matrix. Both matrices have different ways of representing the vertices and edges of a graph. In this paper, the general form of the incidence matrix is determined for the nonnormal subgroup of generalized quaternion groups of order 4푛. The general form is divided into two cases in which when 푛 is odd and when 푛 is even, respectively. The general form is given in the form of block matrices since the graph is disconnected. [ABSTRACT FROM AUTHOR]

A zero divisor graph of a finite ring is defined as a simple graph with its vertices are the zero divisors of the ring, and two distinct vertices are adjacent if and only if their product is equal to the zero element of the ring. In this research, the zero divisor graph is identified for the ring of 2 × 2 matrices over integers modulo prime. Since the ring is noncommutative, the zero divisor graph is a directed graph. First, the vertices of the graph are determined by obtaining the zero divisors of the ring based on its definition. Then, the arcs of the graph are determined by finding the set of annihilators of the ring, which is stated as the set of pairs of elements in the ring where their product is zero. In this research, we define a new term called the square-annihilator. The set of square-annihilators is the set of pairs of the same element where their product is zero. To ensure the zero divisor graph is simple, the elements of the square-annihilators are eliminated from the original set of annihilators. Thus, the general formula for the number of vertices and arcs of the zero divisor graph are obtained. [ABSTRACT FROM AUTHOR]

Let 푅 be a ring. A non-zero divisor graph of 푅 is defined as a graph where its vertices are the non-zero elements of R, and two vertices 푥 and 푦 are adjacent if the product of these two vertices is not equal to zero. In this study, the non- zero divisor graphs are constructed for the ring of Gaussian integers modulo 2푛. The non-zero divisors for the ring need to be found first. Then, the non-zero divisor graph is constructed by using its definition. Two properties of the graph, known as the clique number and the chromatic number are obtained and this graph is found to be not perfect. The total perfect code of the graph is also found in this research. [ABSTRACT FROM AUTHOR]

A Cayley graph of a group is a graph associated with the generating subset of the group. Meanwhile, a pi-Cayley graph is a variation of the Cayley graph where pi is a prime dividing the order of the group. The pi-Cayley graph is focused on the subset that contains the elements with prime power order. In this paper, two graphs are constructed for a cyclic group of order st, namely the s-Cayley graph and the t-Cayley graph where s and t are primes. [ABSTRACT FROM AUTHOR]

The topological index provides information about the overall structure of a molecular graph and is often used in quantitative studies of chemical compounds. Consider Γ be a simple graph, consisting of a collection of edges and vertices. The first Zagreb index of a graph is calculated by adding the squared degrees of every vertex in the graph. Meanwhile, the zero divisor graph of a ring 푅, denoted by Γ(푅), is defined as a graph where its vertices are zero divisors of 푅 and two distinct vertices 푎 and 푏 are adjacent if their product is equal to zero. For 푞 is an odd prime number and 푘 is a positive integer, the first Zagreb index of the zero divisor graph for the commutative ring of integers modulo 2푘푞 is determined in this paper. An example is given to illustrate the main results. [ABSTRACT FROM AUTHOR]

A graph formed by deleting the non-units of a commutative ring R from the complement unit graph, where the units of R represent its vertex set, is referred to as an induced subgraph. A subset C of the vertex set of an induced subgraph of a complement unit graph of R is called a perfect code if the balls of radius 1 centered at C form a partition of the vertex set. In this paper, our focus is on investigating perfect codes within induced subgraphs of complement unit graphs associated with specific commutative rings containing unity. Therefore, by establishing some mathematical theorems, we characterize R for which the induced subgraphs of the complement unit graphs admit either a trivial or a non-trivial perfect code. In addition, some examples are provided as applications of the theorems. [ABSTRACT FROM AUTHOR]

In formal language theory, a grammar is a set of production rules that describes the possible strings in a given language. Context-free grammars, the most applicable type of the grammars, may contain production rules that do not generate any string of the language or affect the string generation processes using several unnecessary steps. Thus, the elimination of such undesirable productions is important for the optimization of the string generation processes. The "cleaning" of context-free grammars is done through the simplification process consisting of a useful substitution rule, removal of useless productions, removal of λ − productions, and removal of unit-productions. This paper presents the simplification of static Watson-Crick context-free grammars; the grammatical counterparts of sticker systems that generate double-stranded strings using context-free rules, several transformations, and substitutions. The simplification process of static Watson-Crick context-free grammars is similar to the simplification process of arbitrary context-free grammars and Watson-Crick context-free grammars. This research shows that for every static Watson-Crick context-free language, there exists an equivalent static Watson-Crick context-free grammar without useless productions, λ − productions, and unit-productions. [ABSTRACT FROM AUTHOR]

Graph splicing is an extended study of splicing on deoxyribonucleic acid (DNA) molecules to counter the intricacy of explaining the idea of DNA splicing in the form of graphs instead of strings. An n-cut splicing is a type of splicing in graph splicing systems, which represents the cleavage pattern of the DNA splicing. An n-cut splicing can be applied to two types of graphs, which are unlabelled and labelled graphs. A labelled graph is a type of graph where the vertices are assigned with symbols or alphabets. After the recombination of the overhangs, the resulted labelled graphs are generated where the set of all resulted labelled graphs is called a language. This language is usually written in the form of strings containing the label of the graph without considering the position of the symbols labelled in the graph, but only the number of the symbols. In computer science, a language can be written as a regular expression where it can generate all strings in the language set. A regular expression varies the output of the generation of the strings and also helps to fix the pattern of the symbols in a specific string. Hence, in this study, the idea of n-cut splicing will be applied to labelled graph with two symbols where the graph can be denoted as a string and the position of each symbol in the graph is known. Besides, the regular expressions of the resulted labelled graph are presented. [ABSTRACT FROM AUTHOR]

The zero-divisor type graph has been introduced to ease the computation of some properties of the zero-divisor graph such as the determination of the graph's perfectness. By extending the idea of the zero-divisor type graph, a new graph namely the nonzero divisor type graph for ring of integers modulo n is introduced in this research. The nonzero divisor type graph for ring of integers modulo n has vertices Td, where d is the nontrivial divisors of n. Two distinct vertices are adjacent if and only if the product of the divisors is not equal to zero. In this paper, two distance-based topological indices which are the Wiener index and mean distance of the nonzero-divisor type graph of the rings of integers modulo paq, for distinct primes p and q and positive integer a are computed. The Wiener index of the graph is the sum of all distances and the mean distance is the average distance between vertices of the graph. [ABSTRACT FROM AUTHOR]

In this study, we introduced a new concept of total non-zero divisor graph of a ring. The total non-zero divisor graph of a ring is defined as a simple undirected graph with its vertices are the non-zero elements of the ring and two distinct vertices are connected if and only if their product is not equal to zero, and their sum is in the its zero divisors sets. In this paper, the total non-zero divisor graph is constructed and the connectivity of the graph is explored. We prove that the total non-zero divisor graph is a null graph for the set of integers modulo p. The connectivity of the total non-zero divisor graph is also determined for the set of integers modulo n, where n ≠ p. [ABSTRACT FROM AUTHOR]

A simple graph is a set of vertices, V(Γ) and a set of edges, E(Γ), where each edge 〈u − v〉 connects two different vertices u and v (there are no self-loops). In topological index, the general zeroth-order Randić index is defined as the sum of the degree of each vertex to the power of α ≠ 0. Given a ring R, let Γ(R) denote the graph whose vertex set is R, such that the distinct vertices a and b are adjacent provided that ab = 0 for the zero-divisor graph of a ring. In this paper, we present the general formula of the general zeroth-order Randić index of the zero-divisor graph for some commutative rings. The commutative ring in the scope of this research is the ring of integers modulo pn, where p is a prime number and n is a positive integer. The general zeroth-order Randić index is found for the cases α =1, 2 and 3. [ABSTRACT FROM AUTHOR]

A semigraph is one of the generalization of graphs. Different from the hypergraph, a semigraph generalizes a graph in a straightforward manner where the edges of the graph are depicted as usual lines as in a graph, instead of sets as in a hypergraph. The vertices of the semigraph can be labelled by assigning symbols to them and the semigraph is called a labelled semigraph. In graph theory, various concepts can be applied on graphs, for instance the splicing system. Splicing system comprises of the field of formal language theory as well as discrete mathematics which focuses on the recombinants behavior of DNA molecules. Labelled semigraph can depict a problem where the labels hold certain values or properties. In DNA computing, a DNA molecule can be illustrated as a labelled semigraph where each vertex is assigned one label representing the nucleotide or base of the DNA molecule; and its cleavage pattern is described as a notion called n-cut splicing. In this research, the idea of n-cut splicing is applied on labelled semigraphs with one symbol. The number of the vertices in a component of n-cut spliced semigraph is firstly determined. Then, the form and order of the new labelled semigraphs generated by the recombination of the labelled n-cut spliced semigraphs are obtained. Besides that, this research also shows that the 180-degree structure of the newly obtained labelled semigraphs is also one of the results of the recombination. [ABSTRACT FROM AUTHOR]

A simple graph is a set of vertices, V(Γ) and a set of edges, E(Γ), where each edge 〈u − v〉 connects two different vertices u and v (there are no self-loops). In topological index, the general zeroth-order Randić index is defined as the sum of the degree of each vertex to the power of α ≠ 0. Given a ring R, let Γ(R) denote the graph whose vertex set is R, such that the distinct vertices a and b are adjacent provided that ab = 0 for the zero-divisor graph of a ring. In this paper, we present the general formula of the general zeroth-order Randić index of the zero-divisor graph for some commutative rings. The commutative ring in the scope of this research is the ring of integers modulo pn, where p is a prime number and n is a positive integer. The general zeroth-order Randić index is found for the cases α =1, 2 and 3. [ABSTRACT FROM AUTHOR]