In this paper, we introduced new index from the Zagreb family, named as -index is defined sum of degree five all vertices a graph. We derive some different graph operations such that Join, Cartesian product, Composition, Corona Tensor Product, Strong Disjunction, Symmetric difference, join Subdivision vertex are obtained.

The Bloch wave functions have been playing a crucial role in the diagnosis of topological phases noninteracting systems. However, waves are no longer applicable presence finite Coulomb interaction and alternative approaches needed to identify indices. In this paper, we focus on three-dimensional higher-order insulators protected by ${C}_{4}T$ symmetry show that index can be computed through eig...

For the inflaton field we determine a new exact solution by using Lie symmetry analysis. Specifically, construct second-order differential master equation for arbitrary scalar potential assuming that spectral index density perturbations $n_{s}$ and to tensor ratio $r$ are related as $n_{s}-1=h\left( r\right) $. Function $h\left( $ is classified according admitted symmetries equation. The possib...

We consider families of theories with large N = 4 superconformal symmetry. We define an index generalizing the elliptic genus of theories with N = 2 symmetry. In contrast to the N = 2 case, the new index constrains part of the non-BPS spectrum. Motivated by aspects of the AdS/CFT correspondence we study the index in the examples of symmetric product theories. We give a physical interpretation o...

We consider families of theories with large N = 4 superconformal symmetry. We define an index generalizing the elliptic genus of theories with N = 2 symmetry. In contrast to the N = 2 case, the new index constrains part of the non-BPS spectrum. Motivated by aspects of the AdS/CFT correspondence we study the index in the examples of symmetric product theories. We give a physical interpretation o...

We consider a 2D isotropic Fermi liquid with attraction in both s and d channels and examine the possibility of a superconducting state with mixed s and d symmetry of the gap function. We show that both in the weak coupling limit and at strong coupling, a mixed s + id symmetry state is realized in a certain range of interaction. Phase transitions between the mixed and the pure symmetry states a...