Topological indices provide important insights into the structural characteristics of molecular graphs. The present investigation proposes and explores a creative graph on a finite group G, which is known as the RIG. This graph is designated as ΓRS G2(4) indicating a simple undirected graph containing elements of G. Two distinct ertices are regarded as nearly the same if and only if their sum yields a non-trivial involution element in G. RIGs have been discovered in various finite groups. We examine several facets of the RIG by altering the graph through the conjugacy classes of G. Furthermore, we investigate the topological indices as applications in graph theory applying the distance matrix of the G2(4) group.

Tobacco products of all kinds are harmful to public health, so legislation has paid great attention to regulating the process of tobacco production and distribution, whether at the level of national or international legislation, in a way that achieves legal protection for these products, so the establishment of civil liability for tobacco companies as a result of harm to the smoker The positive and the negative provoked a jurisprudential dispute due to the specificity of the work of these companies, and the jurisprudence differed in the legal nature of tobacco companies ’liability between contractual and tort liability in a way that enables the injured smoker to obtain his right to compensation.

Background: Mondor's disease means superficial thrombophlibitis of the chest wall in human, treatment is entirely symptomatic. Hot, wet dressing and anodynes may be used for pain relief.
Objective: To evaluate the role of systemic and transdermal action of diclofenac (olfen) with respect to the symptom and sign (pain, erythema along the superficial vein), and the use of Doppler ultrasonography which is a colored ultrasound used for assessment of flow of blood in vessels.
Method: The study was performed on 12 cases with Mondor's disease in middle age female patients with the involvement of lnframammary veins in all of the them (commonly affected), 4 cases had reassurance only, 4 cases had reassurance with systemic diclofenac, and th

     The method of operational matrices is based on the Bernoulli and Shifted Legendre polynomials which is used to solve the Falkner-Skan equation. The nonlinear differential equation converting to a system of nonlinear equations is solved using Mathematica®12, and the approximate solutions are obtained. The efficiency of these methods was studied by calculating the maximum error remainder ( ), and it was found that their efficiency increases as  increases. Moreover, the obtained approximate solutions are compared with the numerical solution obtained by the fourth-order Runge-Kutta method (RK4), which gives  a good agreement.

Voice studies are one of the sensory studies of their first association with auditory taste, which broadcasts its connection to the most sensory point contained in the physical corners, and in order to satisfy our conscience from these fundamental voices, we searched for in modern contexts, high proportions related to the words of Muhammadiyah, so we are looking for (events In the hadiths of the Prophet (peace be upon him), to settle our journey when the true adhkaar received from the Prophet, peace be upon him, and to control us start a journey through which we repeat between the cities of audio repetitions, sometimes we find ourselves have stood at the entrance of what corresponds to the sounds between the title of the Hadith and its t

The group for the multiplication of closets is the set G|N of all closets of N in G, if G is a group and N is a normal subgroup of G. The term “G by N factor group” describes this set. In the quotient group G|N, N is the identity element. In this paper, we procure K(SL(2,125)) and K(SL(2,3125)) from the character table of rational representations for each group.