DEGREE BASED TOPOLOGICAL INDICES OF SANIDIC POLYAMIDES

Mati http://dergipark.gov.tr/mati ISSN:2636-7785 Mati 1 (2) (2019), 64–72. DEGREE BASED TOPOLOGICAL INDICES OF SANIDIC POLYAMIDES Aysun YURTTAS ABSTRACT Sanidic polyamides are special polymers with many applications in textile and clothing industries. In this paper, we calculate several degree based topological graph indices of the sanidic polyamides as these values helps to determine several chemical and physicochemical properties of these polyamides. 1. INTRODUCTION Polyamides are polymers containing repeating amides in the form of ”-CONH-” linkages. The names of the types of polyamides are derived according to the number of carbon atoms in their molecule structures. Some of the naturally occurring polyamides are silk, wool and proteins. Polyamides are classified into two categories. Aliphatic polyamides, known as nylons, and aromatic polyamides, known as aramids. Polyamides find applications in several fields ranging from the textile to the automotive industry. They are used in making medical instruments and clothing, electrical appliances, and in many more areas. Polyamide fibers are used in a wide 2010 Mathematics Subject Classification. 05C07, 05C30, 68R10, 94C15. Keywords and Phrases. Graph, polyamide, sanidic polyamide, topological indices. Received: 05.04.2019 Revised: 28.05.2019 Accepted: 30.05.2019 64 Degree Based Topological Indices of Sanidic Polyamides 65 range of applications due to their excellent mechanical properties and good adhesion property to other materials like rubber. These fibers are used in women’s hosiery and in all the stretch fabrics such as blouses, lingerie, and swimwear. They are also used in several house furnishings such as upholstery and curtains. This type of fibers has mostly been used in some technical textile derivatives like vehicle tires, parachutes, nets and tents. The main factors for preferring polyamide fibers for a wide range of military applications are high strength, elasticity, toughness, and abrasion resistance of them compared with other equivalent materials. In general, polyester has gained considerable significant market share compared to polyamides because of its easy-care characteristics, [7]. In this work, we study on the sanidic polyamide which is one of the aromatic polyamides briefly called aramids, see Fig. 1. In [2], a series of fully aromatic polyesters, polyamides and polyimides having n-alkoxy side chains for 2 ≤ n ≤ 18 have been investigated for their applications in optical microscopy, X-ray analysis and DSC. All members of these series have a rigid backbone and exhibit a decreasing melting range with increasing length of the side chains. This characteristic is very similar to the Wiener index which helps to determine the boiling temperatures of the isomers of alkanes where the longer chains have lower boiling temperature. The polyester with short side chains (2 ≤ n ≤ 6) form nematic melts. Some of aramid applications include the hot-air filtration fabrics, optical-fiber cables, jetengine enclosures, heat-protective clothings, helmets, loudspeaker diaphragms, and reinforced-thermoplastic pipes all having a lot of areas of application. Although the aramids are non-conductive, they are sensitive to UV. They provide good resistance to organic solvents and abrasion which is the main reason to study with them. Figure 1: Sanidic Polyamide where R = C8 H17 . 66 Aysun Yurttas Topological graph indices are defined and used in many areas to study several properties of different objects such as atoms and molecules. A large number of topological graph indices have been defined and studied by many mathematicians and chemists as most graphs are generated from molecules by replacing atoms with vertices and bonds with edges. They are defined as topological graph invariants measuring several physical, chemical, pharmacological, pharmaceutical, biological etc. properties of graphs which are modelling real life situations. They can mainly be grouped into three classes according to the way they are defined: by vertex degrees, by matrices or by distances. Let G = (V, E) be a simple graph with | V (G) |= n vertices and | E(G) |= m edges. That is, no loops nor multiple edges are allowed. For a vertex v ∈ V (G), we denote the degree of v by dG (v) or dv . Two of the most important topological graph indices are called the first and second Zagreb indices denoted by M1 (G) and M2 (G), respectively: (1) M1 (G) = X d2G (u) and M2 (G) = u∈V (G) X dG (u)dG (v). uv∈E(G) They were defined in 1972 by Gutman and Trinajstic, [8], and are referred to due to their uses in QSAR and QSPR studies in chemical studies. In [3], some results on the first Zagreb index together with some other indices are given. For some graph operations, these indices are calculated in [4]. The F -index, also called as forgotten index, of a graph G is denoted by F (G) or M3 (G) and is defined as the sum of the cubes of the degrees of the vertices of the graph. The P total π-electron energy depends on the degree based sums M1 (G) and F (G) = u∈V (G) d3G (u). They were first appeared in the study of structuredependency of total π-electron energy in 1972, [8]. The first index was later named as the first Zagreb index and the second sum has never been further studied until the last few years. As a result, recently, this sum was named as the forgotten index or the F -index briefly by Furtula and Gutman, [6], and it was shown to have an exceptional applicative potential. The hyper-Zagreb index was defined as a variety of the classical Zagreb indices as HM (G) = X (du + dv )2 , (uv∈E see e.g. [6]. Inspired by the study of heat formation for heptanes and octanes in [5], Furtula et. al. proposed an index, called the augmented Zagreb index, which gives Degree Based Topological Indices of Sanidic Polyamides 67 a better prediction power. It is defined by X AZI(G) = uv∈E(G) du dv . du + dv − 2 The harmonic index was introduced by Zhang [11]. It is shown that it correlates well with Π-electron energy of benzenoid hydrocarbons and defined as X H(G) = uv∈E(G) 2 . du + dv Reformulated first, second and third Zagreb indices for a graph G are defined by X ReZG1 (G) = uv∈E(G) X ReZG2 (G) = uv∈E(G) ReZG3 (G) = X du + dv , du · dv du · dv , du + dv (du · dv )(du + dv ). uv∈E(G) Milicevic et. al., [10], reformulated the Zagreb indices in terms of the edge degrees instead of the vertex-degrees as X RM1 (G) = d(e)2 , uv∈E(G) X RM2 (G) = d(e)d(e0 ) e,e0 ∈E(G) where e, e0 are pairs of adjacent edges of the graph G. Aram and Dehgardi, [1], introduced the concept of reformulated F -index as X RF (G) = d(uv)3 . uv∈E(G) Kulli, [9], introduced the first and second Banhatti indices with the intention of taking into account the contributions of pairs of incident elements, not only the vertices or edges. They are defined as X B1 (G) = [dG (u) + d(e)], u,e B2 (G) = X u,e dG (u)d(e). 68 Aysun Yurttas 2. TOPOLOGICAL INDICES OF SANIDIC POLYAM (...truncated)