Open Journal of Discrete Applied Mathematics (ODAM)

The Sombor index has gained lot of attention in the recent days for its mathematical properties and chemical applicabilities. Here, we initiated the novel block number version of the classical Sombor index and its matrix representation of a graph. The Block Sombor index \(BS(G)\) is defined as the sum total of square root of the sum of squares of block numbers of adjacent vertices, where the block number of a vertex is the number of blocks to which that vertex belongs to. The main purpose of this paper is to obtain some bounds and characterizations of \(BS(G)\) and its Block Sombor energy \(E_{BS}\). Also, we estimate some properties of spectral radius of Block Sombor matrix \(A_{BS}(G)\).

We find the maximum and minimum connected unicyclic and connected well-covered unicyclic graphs of a given order with respect to \(\preceq\). This extends 2013 work by Csikv’ari where the maximum and minimum trees of a given order were determined and also answers an open question posed in the same work. Corollaries of our results give the graphs that minimize and maximize \(\xi(G)\) among all connected (well-covered) unicyclic graphs. We also answer more related open questions posed by Oboudi in 2018 and disprove a conjecture due to Levit and Mandrescu from 2008. The independence polynomial of a graph \(G\), denoted \(I(G,x)\), is the generating polynomial for the number of independent sets of each size. The roots of \(I(G,x)\) are called the independence roots of \(G\). It is known that for every graph \(G\), the independence root of smallest modulus, denoted \(\xi(G)\), is real. The relation \(\preceq\) on the set of all graphs is defined as follows, \(H\preceq G\) if and only if \(I(H,x)\ge I(G,x)\text{ for all }x\in [\xi(G),0].\)

In this paper, the fuzzy nonlinear partial differential equations of fractional order are considered. The generalization differential transform method (DTM) and fuzzy variational iteration method (VIM) were applied to solve fuzzy nonlinear partial differential equations of fractional order. The above methods are investigated based on Taylor’s formula, and fuzzy Caputo’s fractional derivative. The proposed methods are also illustrated by some examples. The results reveal the methods are a highly effective scheme for obtaining the fuzzy fractional partial differential equations.

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.