A Study On Unitary Cayley Graphs and Gutman Energy

Abstract

Unitary Cayley Graph is a class of undirected graphs that are highly symmetric with some newlineremarkable properties connecting graph theory and number theory. In this thesis, it is newlinethe keystone which arises with different attributes in the field of algebraic graph theory. newlineFurthermore, we introduce the concept of Gutman energy to facilitate the applications of newlinegraph energy in chemical graph theory. newlineWe investigate the generating functions, namely, Wiener polynomial, hyper-Wiener newlinepolynomials, degree distance polynomial which have substantial applications in chemical newlinegraph theory. However, these applications are by no means restricted to molecular graphs, newlinebut we can also determine a remarkable variety of novel mathematical results. Motivated newlineby this, we compute Wiener polynomial, hyper-Wiener polynomial and degree distance newlinepolynomial of Unitary Cayley Graphs. newlineIn mathematical chemistry, there exist many topological indices which are newlinecounting-related, distance-based, and degree-based. Even though there are such indices, newlinewe restrict here our attention to Gutman index, Harary index, Vertex Padmakar - Ivan index newlineand Edge Padmakar - Ivan index of a simple connected undirected graph. We emphasize the newlinecomputation of these respective indices of Unitary Cayley Graphs. newlineIn 1994, Ivan Gutman introduced the notion of Gutman index. So we intend to newlineintroduce the Gutman matrix and the Gutman eigenvalues of simple connected graph that newlinetraces the concept of Gutman energy. We observe and discuss the Gutman energy of some newlineparticular family of graphs. We include its bounds and applications as well. newlineIn 2012, Chandrashekar Adiga et al., introduced the notion of a novel kind of graph newlineenergy called minimum covering energy, depending on the underlying graph as well as its newlineparticular minimum covering set. This insights assist to mould the concept of minimum newlinecovering Gutman energy, which later bring the notion of minimum covering Gutman energy newlineof Unitary Cayley graphs. Further, we establish the upper and lower bounds for minimum newlinecovering Gutman ener