Open Journal of Discrete Applied Mathematics (ODAM)

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.

In this paper, we define the binomial transform of the generalized fifth order Jacobsthal sequence and as special cases, the binomial transform of the fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas sequences will be introduced. We investigate their properties in details.

The inverse sum indeg index \(ISI(G)\) of a graph is equal to the sum over all edges \(uv\in E(G)\) of weights \(\frac{d_{u}d_{v}}{d_{u}+d_{v}}\). In this paper, we calculated the inverse indeg indices and inverse indeg energies that give information about the physicochemical properties and biological characteristics of Hyaluronic Acid-Paclitaxel conjugates used in the production of drugs used in the treatment of cancer disease. This study presents the relation between the ISI index and the ISI energy of the molecular graph of Hyaluronic Acid-Paclitaxel conjugates.

In this paper, we obtain the quantitative calculation formula of the degree-based topological indices of four standard product for the path and regular graphs, which unify to solve the question on product of these basic graphs without having to deal with it one by one separately. As applications, we give corresponding calculation formula of the general Randić index, the first general Zagreb index and the general sum-connectivity index.