Search Results (17)

A dominating set is a classic concept that is widely used in road safety, disaster rescue operations, and chemical graphs. In this paper, we introduce a variation of the dominating set: the proper 3-dominating set. For a proper 3-dominating set D of graph $G,$ any vertex outside D is adjacent to at least three vertices inside D, and there exists one vertex outside D that is adjacent to three vertices inside D. For graph $G,$ the proper 3-domination number is the minimum cardinality among all proper 3-dominating sets of $G.$ We find that a graph with minimum degree at least 3 or one for which there exists a subgraph with some characteristic always contains a proper 3-dominating set. Further, we find that when certain conditions are met, some graph products, such as the joint product, strong product, lexicographic product, and corona product of two graphs, have a proper 3-dominating set. Moreover, we discover the bounds of the proper 3-domination number. For some special graphs, we get their proper 3-domination numbers. Full article

This work introduces the notion of a hesitant bipolar-valued intuitionistic fuzzy graph (HBVIFG), which reflects four different characterizations: membership with positive/negative aspects and non-membership with positive/negative aspects, incorporating multi-dimensional alternatives in all of its information. HBVIFG generalizes both HBVFG and BVHFG due to its diversified nature in observing four perspectives along with multiple attributes in a piece of information. Numerous studies, examples, and graphical representations emphasize the concept’s distinctiveness and importance. The following graph theory terms are defined: strong directed HBVIFG, full directed HBVIFG, directed spanning HBVIFSG, directed HBVIFSG, and partial directed hesitant bipolar-valued intuitionistic fuzzy subgraph (HBVIFSG). Examples of operations utilizing two HBVIFGs are Cartesian, direct, lexicographical, and strong products. A scenario is used to generate the mapping of relations, which includes homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism. We describe a directed HBVIFG application that employs an algorithm to determine the most dominant person and self-persistent person in a social system and a comparative study is also provided. The proposed method provides a more detailed framework for assessing the most dominant and self-persistent individual in a social network across multi-level attributes along with positive and negative side membership and non-membership grades in each element of a network. Full article

The importance of symmetry in graph theory has always been significant, but in recent years, it has become much more so in a number of subfields, including but not limited to domination theory, topological indices, Gromov hyperbolic graphs, and the metric dimension of graphs. The purpose of this monograph is to initiate the idea of a multi polar q-rung orthopair fuzzy graphs (m-$\mathfrak{PqROPFG})$ as a fusion of multi polar fuzzy graphs and q-rung orthopair fuzzy graphs. Moreover, for a vertex of multi polar q-rung orthopair fuzzy graphs, the degree and total degree of the vertex are defined. Then, some product operations, inclusive of direct, Cartesian, semi strong, strong lexicographic products, and the union of multi polar q-rung orthopair fuzzy graphs (m-$\mathfrak{PqROPFG}s)$, are obtained. Also, at first we define some degree based fuzzy topological indices of m-$\mathfrak{PqROPFG}$. Then, we compute Zareb indices of the first and second kind, Randic indices, and harmonic index of a m-$\mathfrak{PqROPFG}$. Full article

The clustering coefficient of a vertex v, of degree at least 2, in a graph $\Gamma$ is obtained using the formula $C\left(v\right)=\frac{2t\left(v\right)}{\mathrm{deg}\left(v\right)\left(\mathrm{deg}\right(v)-1)},$ where $t\left(v\right)$ denotes the number of triangles of the graph containing v as a vertex, and the clustering coefficient of $\Gamma$ is defined as the average of the clustering coefficient of all vertices of $\Gamma$, that is, $C(\Gamma )=\frac{1}{\left|V\right|}{\sum }_{v\in V}C\left(v\right)$, where V is the vertex set of the graph. In this paper, we give explicit expressions for the clustering coefficient of corona and lexicographic products, as well as for the Cartesian sum; such expressions are given in terms of the order and size of factors, and the degree and number of triangles of vertices in each factor. Full article

The correlations between the physico-chemical properties of a chemical structure and its molecular structure-properties are used in quantitative structure-activity and property relationship studies (QSAR/QSPR) by using graph-theoretical analysis and techniques. It is well known that some structure-activity and quantitative structure-property studies, using eccentric distance sum, are better than the corresponding values obtained by using the Wiener index. In this article, we give precise expressions for the eccentric distance sum polynomial of some graph products such as join, Cartesian, lexicographic, corona and generalized hierarchical products. We implement our outcomes to calculate this polynomial for some significant families of molecular graphs in the form of the above graph products. Full article

A distance antimagic graph is a graph G admitting a bijection $f:V(G)\to \{1,2,\dots ,|V(G)\left|\right\}$ such that for two distinct vertices x and y, $\omega (x)\ne \omega (y)$, where $\omega (x)={\sum }_{y\in N(x)}f(y)$, for $N(x)$ the open neighborhood of x. It was conjectured that a graph G is distance antimagic if and only if G contains no two vertices with the same open neighborhood. In this paper, we study several distance antimagic product graphs. The products under consideration are the three fundamental graph products (Cartesian, strong, direct), the lexicographic product, and the corona product. We investigate the consequence of the non-commutative (or sometimes called non-symmetric) property of the last two products to the antimagicness of the product graphs. Full article

Let G be a graph with vertex set $V\left(G\right)$ and $f:V\left(G\right)\to \{\varnothing ,\{1\},\{2\},\{1,2\left\}\right\}$ be a function. We say that f is an outer-independent 2-rainbow dominating function on G if the following two conditions hold: $\left(\mathrm{i}\right)$$V_{\varnothing }=\{x\in V\left(G\right):f\left(x\right)=\varnothing \}$ is an independent set of G. $\left(\mathrm{ii}\right)$${\cup }_{u\in N\left(v\right)}f\left(u\right)=\{1,2\}$ for every vertex $v\in V_{\varnothing }$. The outer-independent 2-rainbow domination number of G, denoted by ${\gamma }_{r2}^{oi}\left(G\right)$, is the minimum weight $\omega \left(f\right)={\sum }_{x\in V\left(G\right)}\left|f\left(x\right)\right|$ among all outer-independent 2-rainbow dominating functions f on G. In this note, we obtain new results on the previous domination parameter. Some of our results are tight bounds which improve the well-known bounds $\beta \left(G\right)\le {\gamma }_{r2}^{oi}\left(G\right)\le 2\beta \left(G\right)$, where $\beta \left(G\right)$ denotes the vertex cover number of G. Finally, we study the outer-independent 2-rainbow domination number of the join, lexicographic, and corona product graphs. In particular, we show that, for these three product graphs, the parameter achieves equality in the lower bound of the previous inequality chain. Full article

Let G be a graph with no isolated vertex and let $N\left(v\right)$ be the open neighbourhood of $v\in V\left(G\right)$. Let $f:V\left(G\right)\to \{0,1,2\}$ be a function and $V_i=\{v\in V\left(G\right):f\left(v\right)=i\}$ for every $i\in \{0,1,2\}$. We say that f is a strongly total Roman dominating function on G if the subgraph induced by $V_1\cup V_2$ has no isolated vertex and $N\left(v\right)\cap V_2\ne \varnothing$ for every $v\in V\left(G\right)V_2$. The strongly total Roman domination number of G, denoted by ${\gamma }_{tR}^s\left(G\right)$, is defined as the minimum weight $\omega \left(f\right)={\sum }_{x\in V\left(G\right)}f\left(x\right)$ among all strongly total Roman dominating functions f on G. This paper is devoted to the study of the strongly total Roman domination number of a graph and it is a contribution to the Special Issue “Theoretical Computer Science and Discrete Mathematics” of Symmetry. In particular, we show that the theory of strongly total Roman domination is an appropriate framework for investigating the total Roman domination number of lexicographic product graphs. We also obtain tight bounds on this parameter and provide closed formulas for some product graphs. Finally and as a consequence of the study, we prove that the problem of computing ${\gamma }_{tR}^s\left(G\right)$ is NP-hard. Full article

A digraph D is an efficient open domination digraph if there exists a subset S of $V\left(D\right)$ for which the open out-neighborhoods centered in the vertices of S form a partition of $V\left(D\right)$ . In this work we deal with the efficient open domination digraphs among four standard products of digraphs. We present a method for constructing the efficient open domination Cartesian product of digraphs with one fixed factor. In particular, we characterize those for which the first factor has an underlying graph that is a path, a cycle or a star. We also characterize the efficient open domination strong product of digraphs that have factors whose underlying graphs are uni-cyclic graphs. The full characterizations of the efficient open domination direct and lexicographic product of digraphs are also given. Full article

Let G be a simple connected graph. In this paper, we study the spectral properties of the generalized distance matrix of graphs, the convex combination of the symmetric distance matrix $D\left(G\right)$ and diagonal matrix of the vertex transmissions $Tr\left(G\right)$ . We determine the spectrum of the join of two graphs and of the join of a regular graph with another graph, which is the union of two different regular graphs. Moreover, thanks to the symmetry of the matrices involved, we study the generalized distance spectrum of the graphs obtained by generalization of the join graph operation through their eigenvalues of adjacency matrices and some auxiliary matrices. Full article

The picture fuzzy set is an efficient mathematical model to deal with uncertain real life problems, in which a intuitionistic fuzzy set may fail to reveal satisfactory results. Picture fuzzy set is an extension of the classical fuzzy set and intuitionistic fuzzy set. It can work very efficiently in uncertain scenarios which involve more answers to these type: yes, no, abstain and refusal. In this paper, we introduce the idea of the picture fuzzy graph based on the picture fuzzy relation. Some types of picture fuzzy graph such as a regular picture fuzzy graph, strong picture fuzzy graph, complete picture fuzzy graph, and complement picture fuzzy graph are introduced and some properties are also described. The idea of an isomorphic picture fuzzy graph is also introduced in this paper. We also define six operations such as Cartesian product, composition, join, direct product, lexicographic and strong product on picture fuzzy graph. Finally, we describe the utility of the picture fuzzy graph and its application in a social network. Full article

The q-rung orthopair fuzzy graph is an extension of intuitionistic fuzzy graph and Pythagorean fuzzy graph. In this paper, the degree and total degree of a vertex in q-rung orthopair fuzzy graphs are firstly defined. Then, some product operations on q-rung orthopair fuzzy graphs, including direct product, Cartesian product, semi-strong product, strong product, and lexicographic product, are defined. Furthermore, some theorems about the degree and total degree under these product operations are put forward and elaborated with several examples. In particular, these theorems improve the similar results in single-valued neutrosophic graphs and Pythagorean fuzzy graphs. Full article

A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no two adjacent or incident elements receive the same color. The total chromatic number of a graph G, denoted by ${\chi }^{″}\left(G\right)$ , is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G, $\Delta \left(G\right)+1\le {\chi }^{″}\left(G\right)\le \Delta \left(G\right)+2$ , where $\Delta \left(G\right)$ is the maximum degree of G. In this paper, we prove the total coloring conjecture for certain classes of graphs of deleted lexicographic product, line graph and double graph. Full article

Some years ago, the harmonic polynomial was introduced to study the harmonic topological index. Here, using this polynomial, we obtain several properties of the harmonic index of many classical symmetric operations of graphs: Cartesian product, corona product, join, Cartesian sum and lexicographic product. Some upper and lower bounds for the harmonic indices of these operations of graphs, in terms of related indices, are derived from known bounds on the integral of a product on nonnegative convex functions. Besides, we provide an algorithm that computes the harmonic polynomial with complexity $O\left(n^2\right)$ . Full article