Open Journal of Discrete Applied Mathematics (ODAM)

This paper presents several fixed point theorems for intuitionistic generalized fuzzy metric spaces with an implicit relation. Specifically, we utilize compatible maps of type \((\beta)\) in intuitionistic generalized fuzzy metric spaces to derive our fixed point theorems. Our results not only extend but also generalize some fixed point theorems that were previously established in complete fuzzy metric spaces. This is achieved by introducing a novel technique, which enhances the applicability and scope of the existing fixed point theorems.

The paper is concerned with the KG-Sombor index (\(KG\)), a recently introduced vertex-and-edge-degree-based version of the Sombor index, applied to Kragujevac trees (\(Kg\)). A general combinatorial expression for \(KG(Kg)\) is established. The species with minimum and maximum \(KG(Kg)\)-values are determined.

In an improper coloring, an edge $uv$ for which, \(c(u)=c(v)\) is called a bad edge. The notion of the chromatic completion number of a graph \(G\) denoted by \(\zeta(G),\) is the maximum number of edges over all chromatic colorings that can be added to \(G\) without adding a bad edge. We introduce the stability of a graph in respect of chromatic completion. We prove that the set of chromatic completion edges denoted by \(E_\chi(G),\) which corresponds to \(\zeta(G)\) is unique if and only if \(G\) is stable in respect of chromatic completion. After that, chromatic completion and stability regarding Johan coloring are discussed. The difficulty of studying chromatic completion of graph operations is shown by presenting results for two elementary graph operations.

This note establishes the induced vertex stress, total induced vertex stress, vertex stress and total vertex stress of the generalized Johnson graphs of diameter \(2\). The note serves as the foundation to establish the same parameters for generalized Johnson graphs of diameter greater than or equal to \(3\).

The Sombor index (\(SO\)) is a vertex-degree-based graph invariant, defined as the sum over all pairs of adjacent vertices of \(\sqrt{d_i^2+d_j^2}\), where \(d_i\) is the degree of the \(i\)-th vertex. It has been conceived using geometric considerations. Numerous researches of \(SO\) that followed, ignored its geometric origin. We now show that geometry-based reasonings reveal the geometric background of several classical topological indices (Zagreb, Albertson) and lead to a series of new \(SO\)-like degree-based graph invariants.

In 2006, Konstantinova proposed the marginal entropy of a graph based on the Wiener index. In this paper, we obtain the marginal entropy of the complete multipartite graphs, firefly graphs, lollipop graphs, clique-chain graphs, Cartesian product and join of two graphs, which extends the results of ¸Sahin.

The concept of Lucky colorings of a graph is used to introduce the notion of the Lucky \(k\)-polynomials of null graphs. We then give the Lucky \(k\)-polynomials for complete split graphs and generalized star graphs. Finally, further problems of research related to this concept are discussed.

In the present communication, we introduce the theory of Type-II generalized Pythagorean bipolar fuzzy soft sets and define complementation, union, intersection, AND, and OR. The Type-II generalized Pythagorean bipolar fuzzy soft sets are presented as a generalization of soft sets. We showed De Morgan’s laws, associate laws, and distributive laws in Type-II generalized Pythagorean bipolar fuzzy soft set theory. Also, we advocate an algorithm to solve the decision-making problem based on a soft set model.

A \(k\)-plane tree is a tree drawn in the plane such that the vertices are labeled by integers in the set \(\{1,2,\ldots,k\}\), the children of all vertices are ordered, and if \((i,j)\) is an edge in the tree, where \(i\) and \(j\) are labels of adjacent vertices in the tree, then \(i+j\leq k+1\). In this paper, we construct bijections between these trees and the sets of \(k\)-noncrossing increasing trees, locally oriented \((k-1)\)-noncrossing trees, Dyck paths, and some restricted lattice paths.

TEMO = topological effect on molecular orbitals was discovered by Polansky and Zander in 1982, in connection with the eigenvalues of molecular graphs. Eventually, analogous regularities were established for a variety of other topological indices. We now show that a TEMO-type regularity also holds for the Sombor index (\(SO\)): For the graphs \(S\) and \(T\), constructed by connecting a pair of vertex-disjoint graphs by two edges, \(SO(S) < SO(T)\) holds. Analogous relations are verified for several other degree-based graph invariants.