Bounds on the General Eccentric Distance Sum of Graphs

Abstract

A topological graph index is a mathematical formula that can be applied to any graph that models some molecular structure. Topological indices are used to study the quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR)to predict the physicochemical properties of molecules. The eccentric distance sum of a graph G is one of these topological indices and it is defined by EDS(G) = ∑v∈V(G) eccG(v)DG(v), where eccG(v) is the distance between v and a vertex furthest from v in G and DG(v) is the sum of all distances from v to all other vertices in G. What we refer to as a general eccentric distance sum in this dissertation is obtained by raising eccG(v) and DG(v) to powers of a and b (a,b ∈ R), respectively, in the eccentric distance sum of a graph G. The focus of this study is finding bounds on the general eccentric distance sum of graphs. One of the motivations of this study is the great discriminating power of eccentric distance sum of graphs and its application in the study of structure activity/property relationships. For example, the anti-HIV activity of dihy droseselins was explored by using the structure-activity relationship of eccentric distance sum of its associated graph. Motivated by [Vertik, T. (2021), General eccentric distance sum of graphs, Discrete Math ematics, Algorithms, and Applications, Vol. 13, No. 4: 2150046], we present bounds on the general eccentric distance sum for trees with a given bipartition. Moreover, we obtain bounds on the general eccentric distance sum for trees with a given order and specific diameter and domination number. Furthermore, we establish bounds on the general eccentric distance sum for bipartite graphs with a given order and matching number; and a given order and specific diameter. We also establish lower and upper bounds on the general eccentric distance sum for any graph with a given order and diameter; and for a join of two graphs with a given order and number of vertices of maximal degree. Graph transformation operations, proof by contradiction and direct proof techniques are used to obtain the main results in this dissertation.