Let G be a finite group, the result is the involution graph of G, which is an undirected simple graph denoted by the group G as the vertex set and x, y ∈ G adjacent if xy and (xy)2 = 1. In this article, we investigate certain properties of G, the Leech lattice groups HS and McL. The study involves calculating the diameter, the radius, and the girth of ΓGRI.

      Assume that G is a finite group and X is a subset of G. The commuting graph is denoted by С(G,X) and has a set of vertices X with two distinct vertices x, y Î X, being connected together on the condition of xy = yx. In this paper, we investigate the structure of Ϲ(G,X) when G is a particular type of Leech lattice groups, namely Higman–Sims group HS and Janko group J₂, along with  X as a G-conjugacy class of elements of order 3. We will pay particular attention to analyze the discs’ structure and determinate the diameters, girths, and clique number for these graphs.

In this work the poly vinyl imides were prepared as thermal l y stable polymers by modified PVC and PVC copolymers \'lith different cyclic imides such as naphtha imide, phtbal imide, male imide and succin imide as a pendant groups.

This method gave new polyimides, from considered  especially plastic because of their outstanding high  performance properties. As

such, they are priced well above commodity polymers such as PVC

and PVC copolymers.

All the prepared polymers were characterized by IR, UV spectroscopy, elemental analysis and chemical reactions, and thermal analysis such as DTA and TG. Physical properties and the viscosities for all prepared polyimides were determined by using DMF as a sol

With simple and undirected connected graph Φ, the Schultz and modified Schultz polynomials are defined as  and , respectively, where the summation is taken over all unordered pairs of distinct vertices in V(Φ), where V(Φ) is the vertex set of Φ, degu  is the degree of vertex u and d(v,u) is the ordinary distance between v and u, u≠v. In this study, the Shultz distance, modified Schultz distance, the polynomial, index, and average for both have been generalized, and this generalization has been applied  to some special graphs.

In this paper, the -caps are created by action of groups on the three-dimensional projective space over the Galois field of order eight. The types of -caps are also studied and determined either they form complete caps or not.