Volume 2022 Issue 2

Author(s): Ivan Gutman1, Izudin Redžepović1, Veerabhadrappa R. Kulli2
1Faculty of Science, University of Kragujevac, 34000 Kragujevac, Serbia.
2Department of Mathematics, Gulbarga University, Kalaburgi 585 106, India.
Abstract:

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.

Author(s): Ivan Gutman1, Izudin Redžepovi´c1, Veerabhadrappa R. Kulli2
1Faculty of Science, University of Kragujevac, 34000 Kragujevac, Serbia.
2Department of Mathematics, Gulbarga University, Kalaburgi 585 106, India.
Abstract:

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.

Author(s): Eunice Gogo Mphako-Banda1, Johan Kok2
1School of Mathematical Sciences, University of Witwatersrand, Johannesburg, South Africa.
2Independent Mathematics Researcher, City of Tshwane, South Africa & Visiting Faculty at CHRIST (Deemed to be a University), Bangalore, India.
Abstract:

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.

Author(s): Johan Kok1
1Independent Mathematics Researcher, City of Tshwane, South Africa & Visiting Faculty at CHRIST (Deemed to be a University), Bangalore, India.
Abstract:

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 \(3\).

Author(s): Ivan Gutman1
1Faculty of Science, University of Kragujevac, Kragujevac, Serbia.
Abstract:

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.