Mobeen Munir

The spectral radius [ ] of graph [ ] is an essential component that is associated with the eigenvalues of the matrix of graph [ ] and, chemically, with the intermolecular forces. This graph invariant has many useful applications in computer sciences, networking, and molecular computing. There are numerous variants of the [ ] attained by substituting another matrix in place of an adjacency matrix. First Zagreb spectral radius, [ ] is defined as the largest absolute eigenvalue of the first Zagreb matrix and Second Zagreb spectral radius, [ ] is defined as the largest absolute eigenvalue of the second Zagreb matrix. The major focus of this paper is on the [ ], [ ] of the generalized shadow and splitting graphs. The only realistic problem in which we are particularly interested in how [ ] and [ ] are comparable to [ ] and [ ], respectively. We were able to address this challenge by focusing on splitting graphs. We are also interested in how [ ] and [ ] are comparable to [ ] and [ ], respectively. We were able to address this challenge by focusing on shadow graphs.

In this paper, we attempt to discuss the algebras of derivations, Lie derivations, generalized derivations and generalized Lie derivations of generalized quaternion ring [ ] where [ ] is a unital ring with a condition [ ]. We in principal give the results on the decomposition of above mentioned maps of [ ] in the terms of above mentioned maps of [ ] and inner derivation of [ ]. The proofs of the decomposition of Lie derivation of [ ] in terms of Lie derivation and Jordan derivation of [ ] and an inner derivation of [ ] and decomposition of generalized Lie derivation of [ ] in terms of a Lie derivation of [ ] and a left multiplier are given. Moreover, we also compute the matrix representation, admitted by the generalized Lie triple derivations of [ ] with respect to the standard basis.

Irregularity measures tend to describe the complexity of networks. Chemical graph theory is a branch of mathematical chemistry that has a significant impact on the development of the chemical sciences. The study of irregularity indices has recently become one of the most active research areas in chemical graph theory. Irregularity indices help us to examine many chemical and biological properties of chemical structures under study. In this article, we study the irregularity indices of the octahedral and icosahedral networks. These networks are used in crystallography, where the topology and structural aspects are carrying some important facts to determine the properties of large structures theoretically. Our results play an important role in pharmacy, drug design, and many other applied areas. We also compared our results graphically to conclude the irregularity with a change in the parameter of structures.

The sum of the absolute eigenvalues of the adjacency matrix make up graph energy. The greatest absolute eigenvalue of the adjacency matrix is represented by the spectral radius of the graph. Both molecular computing and computer science have uses for graph energies and spectral radii. The Albertson (Alb) energies and spectral radii of generalized splitting and shadow graphs constructed on any regular graph is the main focus of this study. The only thing that may be disputed is the comparison of the (Alb) energies and (Alb) spectral radii of the newly formed graphs to those of the base graph. By concentrating on splitting and shadow graph, we compute new correlations between the Alb energies and spectral radius of the new graph and the prior graph.

Graph energy is defined to be the p-norm of adjacency matrix associated to the graph for p = 1 elaborated as the sum of the absolute eigenvalues of adjacency matrix. The graph’s spectral radius represents the adjacency matrix’s largest absolute eigenvalue. Applications for graph energies and spectral radii can be found in both molecular computing and computer science. On similar lines, Inverse Sum Indeg, (ISI) energies, and (ISI) spectral radii can be constructed. This article’s main focus is the ISI energies, and ISI spectral radii of the generalized splitting and shadow graphs constructed on any regular graph. These graphs can be representation of many physical models like networks, molecules and macromolecules, chains or channels. We actually compute the relations about the ISI energies and ISI spectral radii of the newly created graphs to those of the original graph.