Koszul, Loops, Algebraic and Topological Descriptors

Abstract

Koszul, Loops, Algebraic and Topological Descriptors In this thesis we explore interactions between three well established fields namely Algebra, Combinatorics and Chemistry. Main idea is to solve complicated problems in one field by using comparatively easy approaches of the second. We work it out in three different ways. In the first part, we explore chemical properties of some compounds using combinatorial techniques. In recent days atomic properties of chemical structures are intensively explored using results of graph theory depending upon degrees and distances of the associated graphs. In particular, we work on one of the interesting classes of degree based indices, namely, the Zagreb indices. These indices are very useful in QSPR and QSAR studies. We compute Zagreb index, its co index and multiple index for the topological descriptors of 2-dimensional silicon carbons 𝑆𝑖2𝐶3 − 𝐼𝐼𝐼, 𝑆𝑖𝐶3 − 𝐼𝐼𝐼. On the other hand it is of great importance to explore the irregularity of a graph in Chemistry, Pharmacy and Bio technology. We compute irregularity, total irregularity, variance and irregularity index of some networks including (𝐻𝐻𝐶 − 1), (𝐻𝐻𝐶 − 2), (BSN − 1), (BSN − 2) and nanocone 𝑁𝐶𝑘 [𝑛]. In the second part we focus on connections between Algebra and Combinatorics. We construct a family of Wilson loops of different orders with the help of additive and multiplicative groups. We associate these newly obtained algebraic structures to the graphs through link labeling. As a main result of this session we prove that the Latin square of the Wilson loop is connected with a bipartite graph. Similarly its normal sub loop of the Wilson loop is associated with the class of star graphs. Let 𝑅 = 𝑘[𝑥1, … , 𝑥𝑛] be a polynomial ring in 𝑛 variables. There is one to one correspondence between set of all square free monomial ideals in 𝑅 and set of all simplicial complexes over 𝑛 vertices. The 𝑓- and ℎ-vectors of the simplicial complex ∆ play an important role to describe algebraic properties of the associated square free monomial ideal 𝐼∆. Let ∆ 𝑠𝑢𝑏 be subdivided simplicial complex obtained from ∆. In the last part of this thesis, we discuss the transformation maps sending the ℎ-vectors of the simplicial complex ∆ to the ℎ-vectors of the subdivided simplical complex ∆ 𝑠𝑢𝑏 . In the end we give algebraic applications describing the primary decomposition of an ideal 𝐼𝑃 associated to the partially ordered set = 𝐶𝑖 × 𝐶𝑗 , where 𝐶𝑖 is the chain 1 < . . . < 𝑖. Key Words: Molecular graph, topological descriptor, Quasigroup, loop, normal subloop, Latin square, 𝑓-vector, ℎ-vector, simplicial complex.