Eccentricity based graph invariants of extremal graphs

Abstract

Topological indices, sometimes also recognized as a graph-theoretic invariant, maintain the symmetry of the molecular structures and assign a mathematical language to predict features such as the radius of gyrations, viscosity, boiling points, etc. The topological invariant can be considered as a numeric amount which interconnected with a graph which captures the graph topology and is unchanged under graph automorphism. Nowadays, topological indices have been developing attention due to their significance in the domain of computational chemistry. There are certain crucial categories of topological indices with respect to their specific topological features, like degrees of vertices, distances between vertices, eccentricities of vertices, connectivity, matching number, etc. The main focus of this dissertation is to derive extremal graphs with respect to some eccentricity based indices. We determine the extremal conjugated trees with respect to eccentric connectivity index and also eccentric adjacency index among all n-vertex con jugated trees. We focus on the unicyclic graphs with the largest unicyclic graph with respect to eccentric adjacency index with fixed order and girth. We determine the tree with the largest eccentric adjacency index among all the trees with a fixed diameter. Next we derive the extremal trees with the eccentric connectivity and the eccentric adjacency indices among the trees with a given bipartition size, fixed matching number, fixed in dependence number and fixed domination number. We obtain the graphs with fixed cut edges which have the largest eccentric adjacency index and characterized the extremal graphs. We determine the trees with the smallest and the largest total eccentricity index among the class of trees with p pendent vertices. Furthermore, we define a class of trees with a fixed diameter and investigate the trees with the smallest and the largest total eccentricity index in this class.