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.
